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Mugur Alexandru Acu SUBCLASSES OF α-CONVEX FUNCTIONS ”Lucian Blaga” University Publishing House 2008

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  • Mugur Alexandru Acu

    SUBCLASSES OF α-CONVEX FUNCTIONS

    ”Lucian Blaga” University Publishing House

    2008

  • 2

  • Preface

    The concept of α-convex functions was introduced in

    1969, by the great romanian mathematician Petru T.

    Mocanu, with the aim of making a continuously con-

    nection between the notions of starlike functions and

    convex functions. Taking account of the importance de-

    rived from this connection, the study of variously sub-

    classes of α-convex functions become a pursuit for many

    mathematicians from all over the world.

    The present book contain results of the author (with

    complete proofs), and connected results of other math-

    ematicians (without proofs for efficiency reasons), re-

    garding some subclasses of α-convex convex functions,

    and it is addressed to researchers in the field of Geo-

    3

  • metric Functions Theory, students in mathematics and

    other researchers or students in connected fields such is

    engineering (fluids mechanics).

    4

  • Contents

    Preface 3

    1 Preliminaries 7

    1.1 Univalent functions . . . . . . . . . . . . 7

    1.2 Starlike functions . . . . . . . . . . . . . 15

    1.3 Convex functions . . . . . . . . . . . . . 21

    1.4 α-convex functions . . . . . . . . . . . . 26

    1.5 Differential subordinations.

    Admissible functions method . . . . . . . 31

    1.6 Briot-Bouquet differential subordinations 42

    2 Uniformly starlike and

    uniformly convex functions 45

    5

  • 2.1 Uniformly starlike functions . . . . . . . 45

    2.2 Uniformly convex functions . . . . . . . 49

    3 Subclasses of α-convex functions 66

    3.1 The subclasses UM(α) and UMα . . . . 66

    3.2 The subclass UDn,α(β, γ) . . . . . . . . . 68

    3.3 The subclasses UMα(q) and UDn,α(q) . . 75

    3.4 The subclass Mλ,α(q) . . . . . . . . . . . 83

    3.5 The subclass MLn,α(q) . . . . . . . . . . 94

    3.6 The subclass MLβ,α(q) . . . . . . . . . . 110

    Bibliography 122

    6

  • Chapter 1

    Preliminaries

    1.1 Univalent functions

    Definition 1.1.1 A holomorphic (or meromorphic) func-

    tion which is injective in a domain D, is called univalent

    in D.

    We denote with Hu(D) the set of all univalent func-

    tions in a domain D. In the case D = U

    = {z ∈ C : |z| < 1}, we will denote with Hu(U) theclass of holomorphic and univalent in U . The class of

    all holomorphic functions in a domain D will be denoted

    with H(D).

    7

  • Examples

    1.1.1) If f ∈ Hu(D), g ∈ Hu(E) and f(D) ⊂ Ethen g ◦ f ∈ Hu(D).

    1.1.2) The Koebe function f(z) = z(1−z)2 , z ∈ Uis univalent in U .

    Theorem 1.1.1 [25] If f ∈ Hu(D), then f ′(z) 6= 0 forall z ∈ D.

    We remark that for the function f(z) = ez we have

    f ′(z) 6= 0 for all z ∈ C, but ez = ez+2πi show to us thatthis function it is not univalent. From Theorem 1.1.1 we

    deduce that the univalent functions are also conformal

    mappings.

    We denote with

    H [a, n] = {f ∈ H(U) : f(z) = a + anzn + ....} ,

    A = {f ∈ H(U); f(0) = f ′(0)− 1 = 0}(1.1)

    and with

    S = {f ∈ A; f it is univalent .}(1.2)8

  • We remark that a function f ∈ A will have the fol-lowing series expansion in the unit disk U :

    f(z) = z + a2z2 + ... + anz

    n + ... =(1.3)

    = z +∞∑

    j=2

    ajzj , z ∈ U,

    and S = A ∩ Hu(U) = {f ∈ Hu(U); f with the seriesexpansion (1.3) }.

    We can use the unit disc and the above normalization

    conditions, because them are not restrictions, such it is

    easy to see from the next Theorem:

    Theorem 1.1.2 (Riemann′s Theorem)[25] Let D ⊆C, D 6= C a simple-connected domain, w0 ∈ D and α ∈(−π, π). Then will exist a unique function ϕ ∈ Hu(D)such that ϕ(U) = D, ϕ(0) = w0 and arg ϕ

    ′(0) = α.

    To study in the same time with the class S the mero-

    morphic and univalent functions, will be considered the

    class∑

    of the meromorphic and univalent in U− = C\Ufunctions, having ∞ unique pole and the Laurent series

    9

  • expansion:

    ϕ(ζ) = ζ + α0 +α1ζ

    +α2ζ2

    + ... +αnζn

    + ..., |ζ| > 1.

    A function ϕ from∑

    will verify the normalization

    conditions ϕ(∞) = ∞ and ϕ′(∞) = 1. We will alsodenote by

    E(ϕ) = C\ϕ(U−).

    This set it is a continuum in C and contain at least one

    point. The coefficient α0 from the above series expansion

    is given by

    α0 =1

    2π∫

    0

    ϕ(peiθ)dθ, p > 1.

    We will also use in this book the following notation

    ∑0

    = {ϕ ∈∑

    ; ϕ(ζ) 6= 0, ζ ∈ U−}.

    Remark 1.1.1 Let f ∈ S, f(z) = z + a2z2 + .... Then,the function

    g(z) = f

    (1

    z

    )=

    1

    z−1 + a2z−2 + ...=

    z

    1 + a2z−1 + ...

    10

  • = z − a2 + a3z

    + ... ∈∑

    and g(z) 6= 0 for all z ∈ U−, because f ∈ S has no poles.

    Conversely, if g ∈ ∑,

    g(z) = z + b0 +b1z

    + ...

    and c ∈ C∞\g(U−), then the function

    f(z) =1

    g(1

    z

    )− c

    =z

    1 + (b0 − c)z + ... = z + (c− b0)z2 + ... ∈ S

    This mean that we have a bijection between S and∑

    0.

    Theorem 1.1.3 (Gronwall - Bieberbach) [24], [16]

    Let g be a functions with the Laurent series

    g(z) = z +∞∑

    n=0

    bnz−n, z ∈ U−.(1.4)

    Then g ∈ ∑, then the area

    E(g) = π

    (1−

    ∞∑n=1

    n|bn|2)≥ 0,

    11

  • and thus∞∑

    n=1

    n|bn|2 ≤ 1.

    The equality take place for the function

    gθ(z) = z +eiθ

    z, θ ∈ R.

    The above Theorem it′s the starting point for the next

    results.

    Consequence 1.1.1 Let g ∈ ∑ having the form (1.4).Then |b1| ≤ 1, and the equality take place if and only ifg(z) = z + b0 + e

    2iθ/z, where b0 ∈ C and θ ∈ R.

    Consequence 1.1.2 Let f ∈ S having the form (1.3).Then

    |a3 − a22| ≤ 1.

    More, if f it′s a odd functions, then |a3| ≤ 1, and |a3| =1 if and only if

    f(z) =z

    1 + e2iθz2, θ ∈ R.

    Theorem 1.1.4 [17] If f ∈ S and

    f(z) = z + a2z2 + ..., z ∈ U,

    12

  • then |a2| ≤ 2 and |a2| = 2 if and only if f = Kθ; where

    Kθ(z) =z

    (1− eiθz)2 = z +∞∑

    n=2

    ne(n−1)iθzn, z ∈ U,

    θ ∈ R.

    It′s easy to see that the domain Kθ(U) is the complex

    plane except a radii with the origin in the point −14e−iθ.From the Theorem 1.1.4 it is easy to obtain the fol-

    lowing result:

    Theorem 1.1.5 (Koebe-Bieberbach)[17]

    If f ∈ S and w0 6∈ f(U), then |w0| ≥ 1/4 and|w0| = 1/4 if and only if f = Kθ, where θ is give byw0 = −e−iθ/4.

    The Theorem (1.1.5) has the following geometric in-

    terpretation: the disk U(0; 1/4) it′s the disk, centered in

    the origin, with the maximum radii such that to be cov-

    ered by the image f(U) of the unit disk for all functions

    f ∈ S :U(0; 1/4) =

    f∈Sf(U).

    13

  • We call U(0; 1/4) the Koebe disk of the class S, and

    1/4 will be named the Koebe constant of the class S.

    Theorem 1.1.6 (Covering and distortion

    Theorem)[17] If z ∈ U is a fixed point and r = |z|,then for all f ∈ S the following inequalities holds:

    r

    (1 + r)2≤ |f(z)| ≤ r

    (1− r)2(1.5)1− r

    (1 + r)3≤ |f ′(z)| ≤ 1 + r

    (1− r)3(1.6)1− r1 + r

    ≤∣∣∣∣zf ′(z)f(z)

    ∣∣∣∣ ≤1 + r

    1− r .(1.7)

    The above inequalities are sharp and the equalities

    holds if and only if f = Kθ, for a proper value of the

    parameter θ.

    Remark 1.1.2 Let r1 =r

    (1+r)2 and r2 =r

    (1−r)2 . Then

    the geometric interpretations of the inequalities (1.5)

    are:

    U(0, r1) =⋂

    f∈Sf(U(0, r))

    U(0, r2) =⋃

    f∈Sf(U(0, r))

    14

  • From (1.5) it is easy to see that S is a compact class

    of analytic functions.

    Theorem 1.1.7 (Bieberbach)[17] If f ∈ S and f(z) =z+a2z

    2+ ..., z ∈ U , then |an| ≤ n, n ≥ 2 . The extremalfunctions are Kθ, θ ∈ R.

    The above Theorem was proved, by using the Loewner

    (see [45]) method, in 1984 by the mathematician Louis

    de Branges (see [20]).

    1.2 Starlike functions

    Let f a analytic function in U , f(0) = 0 and

    f(z) 6= 0, z 6= 0. We will denote by Cr the image of thecircle {z ∈ C : |z| = r}, 0 < r < 1, thro′ the function f .

    Definition 1.2.1 We say that Cr it is a starlike curve

    in respect to the origin (or briefly, starlike) if the angle

    ϕ = ϕ(r, θ) = arg f(reiθ) between the radius of f(z), z =

    reiθ and the real positive axis, is a increasing function

    15

  • on θ, when θ increase from 0 to 2π. This condition can

    be express by the following inequality

    ∂ϕ

    ∂θ=

    ∂θarg f(z) > 0, z = reiθ, θ ∈ (0, 2π)(1.8)

    We say that f it is starlike function onto the circle

    {|z| = r} if Cr it is a starlike curve.

    Because f(z) 6= 0, for all z 6= 0 we obtain

    Logf(z) = log |f(z)|+ i arg f(z),

    where z = reiθ. By differentiating with respect to θ and

    using the following equality

    ∂z

    ∂θ=

    ∂reiθ

    ∂θ= rieiθ = iz,

    we obtain

    izf ′(z)f(z)

    =∂

    ∂θlog |f(z)|+ i ∂

    ∂θarg f(z)

    From the above we obtain

    ∂θarg f(z) = Re

    zf ′(z)f(z)

    , z = reiθ(1.9)

    The condition (1.8) can be write in the following form

    Rezf ′(z)f(z)

    > 0, for |z| = r(1.10)

    16

  • which express the necessary and sufficiently condition

    such that a function f to be starlike onto the circle {z ∈C : |z| = r}.

    Because zf′(z)

    f(z) is a harmonic function and the above

    condition hold for |z| = r, we can conclude that theabove condition will hold also for |z| ≤ r. From theabove, we conclude that from f starlike function onto

    the circle {z ∈ C : |z| = r}, we obtain that f willbe starlike onto every circle {z ∈ C : |z| = r′}, where0 < r′ < r.

    Definition 1.2.2 We define the radii of starlikeness for

    the function f by

    r∗(f) = sup{

    r; Rezf ′(z)f(z)

    > 0, |z| ≤ r}

    .(1.11)

    If r∗(f) ≥ 1 we will say that f is a starlike function ontothe unit disk U (or briefly, starlike)

    Remark 1.2.1 1) The equality Rezf ′(z)f(z)

    = 0 can not

    hold for z ∈ U , because the function f will reduce toa constant.

    17

  • 2) The condition Rezf ′(z)f(z)

    > 0, z ∈ U , do not assurethat the function f will be univalent in the unit disk.

    If we impose the additional condition f ′(0) 6= 0, thenthe condition Re

    zf ′(z)f(z)

    > 0 will assure that the func-

    tion f will be univalent in the unit disk and f(U)

    it is a starlike domain (with respect to the origin),

    namely, the segment [0, w] ∈ f(U) for all w ∈ f(U).

    Theorem 1.2.1 [43] Let f be a analytic function in U ,

    with f(0) = 0. Then f is univalent in U , and f(U) is a

    starlike domain (with respect to the origin) if and only

    if f ′(0) 6= 0 and

    Rezf ′(z)f(z)

    > 0, for all z ∈ U(1.12)

    Definition 1.2.3 Let denote by S∗ the class of func-

    tions analytic in U, with f(0) = 0, f ′(0) = 1 and which

    are starlike (with respect to the origin) in U , namely

    S∗ = {f ∈ H(U); f(0) = f ′(0)− 1 = 0,

    Rezf ′(z)f(z)

    > 0, z ∈ U}.

    18

  • Remark 1.2.2 By using the subordination, the class S∗

    may be defined: if f(z) = z + a2z2 + ..., z ∈ U , then

    f ∈ S∗ if and only if zf′(z)

    f(z)≺ 1 + z

    1− z , z ∈ U .

    Because the Koebe function Kθ(z) =z

    (1 + eiθz)2, θ ∈

    R is starlike (for a proper value of the parameter θ), we

    conclude that the distortion theorem for the class S hold

    also for the class S∗:

    Theorem 1.2.2 [43] If f ∈ S∗, then the following in-equalities are sharp:

    r

    (1 + r)2≤ |f(z)| ≤ r

    (1− r)2(1.13)1− r

    (1 + r)3≤ |f ′(z)| ≤ 1 + r

    (1− r)3(1.14)1− r1 + r

    ≤∣∣∣∣zf ′(z)f(z)

    ∣∣∣∣ ≤1 + r

    1− r(1.15)

    where z ∈ U, |z| = r, and the extremal function is theKoebe function f = Kθ (for a proper value of the pa-

    rameter θ).

    From the above theorem, we conclude that the class

    S∗ is a compact set.

    19

  • Let

    M [a, b] = {µ : [a, b] → R+,(1.16)

    where µ it is a increasing function

    onto [a, b] ,

    b∫

    a

    dµ(t) = µ(b)− µ(a) = 1}

    Theorem 1.2.3 [43] The function f(z) = z + a2z2 +

    ..., z ∈ U belong to the class S∗ if and only if there exista function µ ∈ M [0, 2π] such that

    f(z) = z exp

    −2

    2π∫

    0

    log(1− ze−it)dµ(t) , z ∈ U.

    Two important subclasses of the class S∗ are the sub-

    class of starlike functions of order α(0 ≤ α < 1), denotedby S∗(α) and the subclass of strongly starlike of order

    α(0 < α ≤ 1), denoted by S∗[α].

    Definition 1.2.4 The function f ∈ A is called starlikeof order α, 0 ≤ α < 1, if the following inequality hold

    Rezf ′(z)f(z)

    > α, z ∈ U.(1.17)

    We denote this class by S∗(α).

    20

  • Definition 1.2.5 The function f ∈ A is calledstrongly starlike of order α, 0 < α ≤ 1 if the followinginequality hold

    ∣∣∣∣argzf ′(z)f(z)

    ∣∣∣∣ < απ

    2, z ∈ U.(1.18)

    We denote this class by S∗[α].

    It is easy to see that S∗(0) = S∗ and S∗[1] = S∗.

    1.3 Convex functions

    Let f a analytic function in U , with f ′(z) 6= 0, for all0 < |z| < 1. Let Cr be the image of the circle {z ∈ C :|z| = r}, 0 < r < 1, by using the function f .

    Definition 1.3.1 The curve Cr is called convex if the

    angle

    ψ(r, θ) =π

    2+ arg zf ′(z), z = reiθ

    between the tangent, into the point f(z), to the curve

    Cr and the real positive axis, is a increasing function on

    θ ∈ [0, 2π] .

    21

  • Definition 1.3.2 The function f is called convex onto

    the circle |z| = r if Cr is a convex curve.The function f is convex onto the circle {z ∈ C :

    |z| = r} if and only if

    Re

    {1 +

    zf ′′(z)f ′(z)

    }> 0, |z| = r.(1.19)

    From the above we obtain that for f convex onto the

    circle

    {z ∈ C : |z| = r}, we have f convex onto every circle{z ∈ C : |z| = r′}, where 0 < r′ < r.

    Definition 1.3.3 We define the radii of convexity for

    the function f by

    rc(f) =(1.20)

    sup

    {r; Re

    {zf ′′(z)f ′(z)

    + 1

    }> 0, |z| ≤ r

    }.

    If rc(f) ≥ 1, we will say that the function f is convexin the unit disk U (or briefly, convex), and f will verify

    the condition

    Re

    {1 +

    zf ′′(z)f ′(z)

    }> 0, |z| < 1.(1.21)

    22

  • The condition(1.21) imply f ′(z) 6= 0, for all0 < |z| < 1.

    Remark 1.3.1 The condition Re

    {1 +

    zf ′′(z)f ′(z)

    }> 0, z ∈

    U do not assure that the function f is univalent in the

    unit disk U (for example the function f(z) = z2 verify

    the above condition, but it is not univalent in U).

    Theorem 1.3.1 [43] Let f be a analytic function in U .

    the function f is univalent in U , and f(U) is a convex

    domain, if and only if f ′(0) 6= 0 and

    Re

    {1 +

    zf ′′(z)f ′(z)

    }> 0, for all z ∈ U(1.22)

    Definition 1.3.4 We will denote by Sc (or by K) the

    class of all analytic functions in U, with f(0) = 0,

    f ′(0) = 1 and which are convex in U , namely

    Sc = {f ∈ H(U); f(0) = f ′(0)− 1 = 0,

    Re

    {1 +

    zf ′′(z)f ′(z)

    }> 0, z ∈ U}

    and Sc ⊂ S.23

  • The connection between the classes S∗ and Sc is es-

    tablish by the following theorem:

    Theorem 1.3.2 [13] Let f ∈ A and g(z) = zf ′(z).Then f ∈ Sc if and only if g ∈ S∗.

    Let consider the integral operator IA : A → A, f =IA(g), g ∈ A, where

    f(z) =

    z∫

    0

    g(t)

    tdt, z ∈ U.(1.23)

    The integral operator IA is called Alexander′s opera-

    tor. By using this operator, the above theorem can be

    express in the following form: Sc = IA(S∗), and IA is a

    bijection between the classes S∗ and Sc.

    Between the classes S∗ and Sc can also be establish

    the following connection:

    Theorem 1.3.3 (Marx and Strohhäcker)[37], [52]

    If f ∈ A, then

    Re

    {1 +

    zf ′′(z)f ′(z)

    }> 0, z ∈ U(1.24)

    24

  • ⇒ Rezf′(z)

    f(z)>

    1

    2, z ∈ U

    We conclude that Sc ⊂ S∗(1/2).

    Theorem 1.3.4 [43] If f(z) = z + a2z2 + a3z

    3 + ...

    belong to the class Sc, then |an| ≤ 1, for all n ≥ 2. Theequality hold if and only if the function f have the form

    f(z) = z1 + eiτz

    , τ ∈ R, z ∈ U .

    Theorem 1.3.5 [43] If f ∈ Sc, then the following in-equalities are sharp:

    r

    1 + r≤ |f(z)| ≤ r

    1− r(1.25)

    1

    (1 + r)2≤ |f ′(z)| 1

    (1− r)2(1.26)

    where z ∈ U, |z| = r. The equalities holds for the func-tion f(z) =

    z

    1 + eiτz, τ ∈ R, z ∈ U , where τ is properly

    choose.

    From (1.25) we conclude that the class Sc is a compact

    set.

    25

  • Remark 1.3.2 Letting r → 1 in (1.25) we find that theKoebe constant for the class Sc is 1/2.

    Definition 1.3.5 We say that the function f ∈ A isconvex of order α, 0 ≤ α, < 1, if the following inequalityhold

    Re

    {1 +

    zf ′′(z)f ′(z)

    }> α, z ∈ U(1.27)

    We denote by Sc(α) the class of all this functions.

    1.4 α-convex functions

    Let f be a analytic function in U , with f(0) = 0,

    f(z)f ′(z)z 6= 0, and let α be a fixed real number.Let χ(r, θ) = (1− α)ϕ(r, θ) + αψ(r, θ).

    Definition 1.4.1 The curve Cr is called α-convex if the

    function χ(r, θ) is a increasing function on the parame-

    ter θ, where θ ∈ [0, 2π], namely∂χ(r, θ)

    ∂θ=(1.28)

    (1− α)∂ϕ(r, θ)∂θ

    + α∂ψ(r, θ)

    ∂θ> 0,

    26

  • where θ ∈ [0, 2π].

    The function f is called α-convex onto the circle

    {z ∈ C; |z| = r} if Cr is a α-convex curve.The function f is α-convex onto the circle

    {z ∈ C; |z| = r} if and only if

    ReJ(α, f ; z) > 0, |z| = r,(1.29)

    where

    J(α, f ; z) =(1.30)

    (1− α)zf′(z)

    f(z)+ α

    (1 +

    zf ′′(z)f ′(z)

    ).

    Taking into the account the properties of the har-

    monic functions, from f is a α-convex function onto the

    circle {z ∈ C; |z| = r}, we conclude that F is a α-convex function onto every circle {z ∈ C; |z| = r′},where 0 < r′ < r.

    Definition 1.4.2 We define the radii of α-convexity for

    the function f by

    rα(f) = sup{r; ReJ(α, f ; z) > 0, |z| ≤ r}.27

  • If rα(f) ≥ 1 we say that the function f is α-convex inthe unit disk U , and f will verify the following condition

    ReJ(α, f ; z) > 0, z ∈ U.(1.31)

    Definition 1.4.3 Let denote by Mα the class of analytic

    functions in U , with f(0) = 0, f ′(0) = 1 and which are

    α-convex in U , namely

    Mα = {f ∈ H(U), f(0) = f ′(0)− 1 = 0,

    ReJ(α, f ; z) > 0, z ∈ U}

    We remark that M0 = S∗ and M1 = Sc.

    Remark 1.4.1 1. By taking p(z) = zf′(z)

    f(z) we obtain

    J(α, f ; z) = p(z) + αzp′(z)p(z)

    ,

    and thus (1.31) can be write in the following form

    Re

    [p(z) + α

    zp′(z)p(z)

    ]> 0, z ∈ U(1.32)

    or

    p(z) + αzp′(z)p(z)

    ≺ 1 + z1− z .(1.33)

    28

  • 2. If the condition (1.32) hold, then p(z) =zf ′(z)f(z)

    is

    analytic in U and p(z) 6= 0, z ∈ U . We con-clude that the condition

    f(z)f ′(z)z

    6= 0, z ∈ U willalso hold.

    Theorem 1.4.1 [43] For α, β ∈ R with0 ≤ β/α ≤ 1, we have Mα ⊂ Mβ.

    Corollarly 1.4.1 For all α ∈ [0, 1], we haveSc ⊂ Mα ⊂ S∗.

    Theorem 1.4.2 [43] If α > 0, then f ∈ Mα if and onlyif F ∈ S∗, where

    F (z) = f(z) =

    [zf ′(z)f(z)

    ]α.

    From the above theorem it is easy to obtain the fol-

    lowing result:

    Theorem 1.4.3 [43] If α > 0, then f ∈ Mα if and onlyif there exist a function F ∈ S∗ such that

    f(z) =

    1

    α

    z∫

    0

    F 1/α(ζ)

    ζdζ

    α

    , z ∈ U.(1.34)

    29

  • Definition 1.4.4 A function f ∈ Mα is called α-convexof order γ, 0 ≤ γ < 1, if

    ReJ(α, f ; z) > γ, z ∈ U.(1.35)

    We denote by Mα(γ) the class of all this functions.

    Concerning the radii of α-convexity for the class S, in

    1972 V.V. Cernikov give the following result:

    Theorem 1.4.4 [19] If coth 1 − 1 = 0.313... ≤ α ≤ 1,then rα(S) = 1 + α−

    √α(α + 2) .

    In 1974 S.S. Miller, P.T. Mocanu and M.O. Reade,

    prove in [42] that the above result hold also for α > 1.

    Theorem 1.4.5 [42] The radii of α-convexity for the

    class S∗ is

    rα(S∗) =

    1 + α−√

    α(α + 2), α ≥ 0√2−√−α2 +

    √−α, −3 < α < 0

    −(1 + α)−√

    α(α + 2), α ≤ −3.

    30

  • 1.5 Differential subordinations.

    Admissible functions method

    Definition 1.5.1 Let f and g be analytic functions in

    U . We say that the function f is subordinate to the

    function g, if there exist a function w, which is analytic

    in U, w(0) = 0, |w(z)| < 1, z ∈ U , such that

    f(z) = g(w(z)), (∀)z ∈ U.

    We denote by ≺ the subordination relation.

    Theorem 1.5.1 [43] Let f be a analytic function in U

    and g be a analytic and univalent function in U . Then

    f ≺ g if and only if f(0) = g(0) and f(U) ⊆ g(U).

    1.4.2 Subordination′s Principle Let f be a an-

    alytic function in U and g be a analytic and univalent

    function in U . Then f(0) = g(0) and f(U) ⊆ g(U) im-plies f(Ur) ⊆ g(Ur), r ∈ (0, 1]. The equality f(Ur) =g(Ur) hold for r < 1 if and only if f(U) = g(U), or

    f(z) = g(eiθz), z ∈ U, θ ∈ R.31

  • The differential subordinations method (also called

    admissible functions method) was introduced by the P.T.

    Mocanu and S.S. Miller in [38] and [39].

    Let Ω , ∆ ∈ C, p be a analytic function in U, withp(0) = a, and let ψ(r, s, t; z) : C3 × U → C.

    Let consider the following implication:

    (1.36)

    {ψ(p(z), zp′(z), z2p′′(z); z) |z ∈ U} ⊂ Ω ⇒ p(U) ⊂ ∆.

    Concerning the above implication we can consider the

    following three problems:

    • Problem 1. We know the sets Ω and ∆, and wewant to find conditions on the function ψ such that

    the implication (1.36) hold. A function ψ, which

    is a solution of the above problem, it is called a

    admissible function.

    • Problem 2. We know the set Ω and the functionψ, and we want to find the set ∆ such that the

    32

  • implication (1.36) hold. If it is possible, we want to

    find the smallest ∆, which is a solution of the above

    problem.

    • Problem 3. We know the set ∆ and the functionψ, and we want to find the set Ω such that the impli-

    cation (1.36) hold. If it is possible, we want to find

    the greatest set Ω such that the implication (1.36)

    hold.

    If ∆ is a simple-connected domain, with a ∈ ∆ and∆ 6= C, then there exist a conformal mapping q fromU to ∆ and such that q(0) = a. In this conditions the

    implication (1.36) can be write in the following form:

    (1.37)

    {ψ(p(z), zp′(z), z2p′′(z); z) | z ∈ U} ⊂ Ω ⇒p(z) ≺ q(z).

    If Ω it is also a simple connected domain and Ω 6= C,then there exist a conformal mapping h from U to Ω

    and such that h(0) = ψ(a, 0, 0; 0).

    33

  • More, if ψ(p(z), zp′(z), z2p′′(z); z) is a analytic func-

    tion in U , then the implication (1.36) can be write in

    the following form:

    ψ(p(z), zp′(z), z2p′′(z); z) ≺ h(z) ⇒(1.38)

    p(z) ≺ q(z).

    From the above we derive the following definitions:

    Definition 1.5.2 Let ψ : C3 × U → C and h be a uni-valent function in U . If p is a analytic function in U

    which satisfy the following differential subordination:

    ψ(p(z), zp′(z), z2p′′(z); z) ≺ h(z),(1.39)

    then p is called a solution of the differential subordina-

    tion.

    Definition 1.5.3 A univalent function q is called a dom-

    inant of the differential subordination (1.39) if p ≺ q, forevery p which satisfy (1.39).

    Definition 1.5.4 A dominant q̃ which satisfy q̃ ≺ q forevery dominant q of the differential subordination (1.39)

    34

  • is called the best dominant of the differential subordina-

    tion (1.39). We remark that the best dominant it is

    unique except a rotation of U .

    If Ω and ∆ are sample connected domains, the above

    three problems can be write in the following forms:

    Problem 1′. Let consider the univalent functions h

    and q. We want to find the class of admissible functions

    ψ[h, q] such that the differential subordination (1.38)

    hold.

    Problem 2′. Let consider the differential subordi-

    nation (1.38). We want to find the dominant q of this

    subordination. If it is possible, we want to find the best

    dominant.

    Problem 3′. Let consider the class ψ and the domi-

    nant q. We want to find the greater class of functions h

    such that the differential subordination (1.38) hold.

    Fundamental lemmas:

    For z0 = r0eiθ0, 0 < r0 < 1, we will denote by Ur0 =

    35

  • {z ∈ C; |z| < r0} the disk with the center into the originand with the radii r0, U r0 =

    {z ∈ C; |z| ≤ r0}.

    Lemma 1.5.1 Let z0 ∈ U , r0 = |z0| and f(z) = anzn++an+1z

    n+1 + ... a continuous function into U r0 and an-

    alytic onto Ur0 ∪ {z0} with f(z) 6≡ 0 and n ≥ 1. If

    |f(z0)| = max{|f(z)|; z ∈ U r0}(1.40)

    then there exist a number m ≥ n ≥ 1 such that(i)

    z0f′(z0)

    f(z0)= m, and

    (ii) Re

    {1 +

    z0f′′(z0)

    f ′(z0)

    }≥ m.

    A particular version (z0 = f(z0) = 1) of the first item

    of the above lemma was considered in 1925 like a open

    problem by K. Loewner. The form presented above, was

    considered in 1971 by I.S. Jack.

    Definition 1.5.5 Let Q be the class of all analytical and

    injective functions q defined onto U\E(q), where

    E(q) =

    {ζ ∈ ∂U ; lim

    z→ζq(z) = ∞

    },

    36

  • and q′(ζ) 6= 0 for every ζ ∈ ∂U\E(q).

    Lemma 1.5.2 Let q ∈ Q cu q(0) = a and p(z) = a +pnz

    n + ... be a analytic function in U , with p(z) 6≡ a andn ≥ 1. If there exist z0 ∈ U and ζ0 ∈ ∂U\E(q) such thatp(z0) = q(ζ0) and p(Ur0) ⊂ q(U), where r0 = |z0|, thenthere exist a number m ≥ n such that:

    (i) z0p′(z0) = mζ0q′(ζ0) and

    (ii) Re

    {z0p

    ′′(z0)p′(z0)

    + 1

    }≥ mRe

    {ζ0q

    ′′(ζ0)q′(ζ0)

    + 1

    }.

    Lemma 1.5.3 Let q ∈ Q, with q(0) = a, and let p(z) =a + pnz

    n + ... be a analytic function in U with p(z) 6≡ aand n ≥ 1. If p 6≺ q then there exist z0 = r0eiθ0 ∈ U andζ0 ∈ ∂U\E(q) and a number m ≥ n ≥ 1 such that

    (i) p(Ur0) ⊂ q(U),(ii) p(z0) = q(ζ0),

    (iii) z0p′(z0) = mζ0q′(ζ0) and

    (iv) Re

    {z0p

    ′′(z0)p′(z0)

    + 1

    }≥ Re

    {ζ0q

    ′′(ζ0)q′(ζ0)

    + 1

    }.

    Definition 1.5.6 Let Ω be a set from C, q ∈ Q and n ∈N. The class of admissible functions ψn[Ω, q] contain all

    37

  • the functions ψ : C3×U → C which satisfy the followingadmissibility condition care

    ψ(r, s, t; z) 6∈ Ω for r = q(ζ), s = mζq′(ζ),Re

    [ts + 1

    ] ≥ mRe[

    ζq′′(ζ)q′(ζ) + 1

    ],

    z ∈ U, ζ ∈ ∂U\E(q) and m ≥ n.(1.41)

    We will also use the following notation ψ1[Ω, q] =

    ψ[Ω, q].

    For Ω a sample connected domain, Ω 6= C and h bea conformal mapping from U to Ω, we will denote the

    class of admissible functions by ψn[h, q].

    If ψ : C2 × U → C, then the admissibility condition(1.41) become

    ψ(q(ζ),mζq′(ζ); z) 6∈ Ω,

    where z ∈ U, ζ ∈ ∂U\E(q) and m ≥ n.If Ω ⊂ Ω̃, then ψn(Ω̃, q) ⊂ ψn(Ω, q) and

    ψn[Ω, q] ⊂ ψn+1[Ω, q].

    Theorem 1.5.2 [43] Let ψ ∈ ψn[Ω, q], with q(0) = a.If p(z) = a+pnz

    n+... is a analytic function in U , which

    38

  • satisfy the following condition

    ψ(p(z), zp′(z), z2p′′(z); z) ∈ Ω, z ∈ U ,(1.42)

    then p ≺ q.

    Corollarly 1.5.1 Let Ω ⊂ C and q a univalent functionin U with q(0) = a. Also, let ψ ∈ ψn[Ω, qρ], ρ ∈ (0, 1),where qρ(z) = q(ρz).

    If p(z) = a+pnzn + ... is a analytic function in U , which

    satisfy the following condition

    ψ(p(z), zp′(z), z2p′′(z); z) ∈ Ω, z ∈ U ,

    then p ≺ q.

    Let consider Ω 6= C a sample connected domain.

    Theorem 1.5.3 [43] Let ψ ∈ ψn[h, q] with q(0) = aand ψ(a, 0, 0; 0) = = h(0). If p(z) = a + pnz

    n + ... and

    ψ(p(z), zp′(z), z2p′′(z); z) is analytic in U and satisfy the

    following condition

    ψ(p(z), zp′(z), z2p′′(z); z) ≺ h(z) ,(1.43)

    then p ≺ q.39

  • Corollarly 1.5.2 Let h , q be two univalent functions

    in U and q(0) = a. Also, let ψ : C3 × U → C, withψ(a, 0, 0; 0) = h(0), which satisfy one from the following

    conditions:

    (i) ψ ∈ ψn[h, qρ], for a number ρ ∈ (0, 1)(ii) there exist a number ρ0 ∈ (0, 1) such that ψ ∈

    ψn[hρ, qρ] for every ρ ∈ (ρ0, 1), where qρ(z) = q(ρz) andhρ(z) = h(ρz)

    If p(z) = a + pnzn + ..., ψ(p(z), zp′(z), z2p′′(z); z) is

    analytic in U and

    ψ(p(z), zp′(z), z2p′′(z); z) ≺ h(z),

    then p ≺ q.

    The following theorems will refer to the best dominant

    of the differential subordination (1.43):

    Theorem 1.5.4 [43] Let h be a univalent function in

    U and ψ : C3×U → C. Let assume that the differentialequation

    ψ(p(z), zp′, z2p′′(z); z) = h(z)(1.44)

    40

  • has a solution q which satisfy one from the following

    conditions:

    (i) q ∈ Q and ψ ∈ ψ[h, q] or

    (ii) q is univalent in U and ψ ∈ ψ[h, qρ], for a numberρ ∈ (0, 1) or

    (iii) q is univalent in U and there exist a number

    ρ0 ∈ (0, 1) such that ψ ∈ ψn[hρ, qρ] for every ρ ∈ (ρ0, 1).

    If p(z) = q(0) + p1z + ..., ψ(p(z), zp′, z2p′′(z); z) is

    analytic in U , and p satisfy the condition (1.43), then

    p ≺ q and q is the best dominant.

    Theorem 1.5.5 [43] Let ψ ∈ ψn[Ω, q] and f a analyticfunction in U with f(U) ⊂ Ω. If the differential equation

    ψ(p(z), zp′, z2p′′(z); z) = f(z)

    has a solution p(z) which is analytic in U , with p(0) =

    q(0), then p ≺ q.

    41

  • 1.6 Briot-Bouquet differential subordinations

    Definition 1.6.1 Let β and γ be two complex numbers,

    h a univalent function in U and p(z) = h(0)+p1z + ... a

    analytic function in U which satisfy the subordination:

    p(z) +zp′(z)

    βp(z) + γ≺ h(z).(1.45)

    This first order differential subordination is called a

    Briot-Bouquet differential subordination.

    By using the differential subordinations method, in

    [40] and [41], are obtained many usefully result regarding

    the Briot-Bouquet differential subordinations or regard-

    ing generalizations of Briot-Bouquet differential subor-

    dinations.

    Theorem 1.6.1 [40], [41] Let h be a convex function

    in U such that Re[βh(z) + γ] > 0, z ∈ U . If p is aanalytic function in U , with p(0) = h(0), and p satisfy

    the Briot-Bouquet differential subordination (1.45), then

    p ≺ h.

    42

  • Theorem 1.6.2 [40], [41] Let h be a convex function

    in U and assume that the differential equation

    q(z) +zq′(z)

    βq(z) + γ= h(z), q(0) = h(0)(1.46)

    has a univalent solution which satisfy the subordination

    q ≺ h. If p is a analytic function in U and satisfythe Briot-Bouquet differential subordination (1.45), then

    p ≺ q ≺ h and q is the best dominant.

    Remark 1.6.1 By using the previously result, the proof

    of the Marx and Strohäcker Theorem′s (1.3.3) become a

    sample verification (with h(z) =

    (1 + z)/(1− z), q(z) = 1/(1− z), β = 1 and γ = 0).

    Theorem 1.6.3 [40], [41] Let h be a convex function

    in U such that Re[βh(z) + γ] > 0, z ∈ U and assumethat the differential equation

    q(z) +zq′(z)

    βq(z) + γ= h(z), q(0) = h(0)

    has a univalent solution q. If p is a analytic function

    in U which satisfy the Briot-Bouquet differential sub-

    43

  • ordination (1.45), then p ≺ q ≺ h and q is the bestdominant.

    Theorem 1.6.4 [40], [41] Let q be a convex function

    in U and j : U → C with Re[j(z)] > 0.If p ∈ H(U), which satisfy the subordinationp(z) + j(z) · zp′(z) ≺ q(z), then p ≺ q.

    44

  • Chapter 2

    Uniformly starlike and

    uniformly convex functions

    2.1 Uniformly starlike functions

    The notion of uniformly starlike function was introduced

    in 1991 by A.W. Goodman (see [22]) and was inspired

    by the following open problem:

    Pinchuk′s problem. Let f ∈ S∗ and let γ be acircle contained in U with the center ζ also in U . It is

    f(γ) a closed starlike curve with respect to f(ζ)?

    Because the above problem has a negative answer,

    45

  • A.W. Goodman consider a more ”strongly” condition in

    the definition of the uniformly starlike functions:

    Definition 2.1.1 A function f is called uniformly star-

    like in U if f ∈ S∗ and for any circular arc γ from U ,with the center ζ also in U , the arc f(γ) is starlike with

    respect to f(ζ). We denote by US∗ the class of all this

    functions.

    In [22] Goodman prove that for the arc γ = z(t), we

    have the arc f(γ) starlike with respect to ω0, if and only

    if

    Im

    [f ′(z)

    f(z)− ω0 ·dz

    dt

    ]≥ 0,(2.1)

    for z onto γ. For γ = Cr = {z ∈ C; |z| = r} and ω0 = 0it is easy to see that we obtain the condition (1.10).

    For z = ζ + reit we have z′(t) = i(z − ζ), and weobtain:

    Theorem 2.1.1 [22] Let f ∈ S. Then f ∈ US∗ if andonly if

    Ref(z)− f(ζ)(z − ζ)f ′(z) > 0,(2.2)

    46

  • for any (z, ζ) from U × U .

    Theorem 2.1.2 [22] The function

    f1(z) =z

    1− Az = z +∞∑

    n=2

    An−1zn, z ∈ U(2.3)

    belong to the class US∗ if and only if |A| ≤ 1√2 '0, 7071.

    Theorem 2.1.3 [22] If

    f2(z) = z + Azn, n ≥ 1, z ∈ U(2.4)

    and |A| ≤ √2/(2n), then f2 belong to the class US∗.

    Remark 2.1.1 If f ∈ US∗, then for ζ = −z we obtain

    Re2zf ′(z)

    f(z)− f(−z) ≥ 0, z ∈ U.(2.5)

    A function f ∈ A which verify the condition (2.5) itis called a starlike function with respect to symmetric

    points. This class of functions was introduced by Sak-

    aguci in [49].

    Theorem 2.1.4 [22] If f ∈ US∗ and |z| = r < 1, then:r

    1 + 2r≤ |f(z)| ≤ −r + 2 ln 1

    1− r .(2.6)

    47

  • Theorem 2.1.5 [22] Let f ∈ S, f(z) = z +∞∑

    n=2anz

    n. If

    ∞∑n=2

    n|an| ≤√

    2/2,(2.7)

    then f ∈ US∗.

    Definition 2.1.2 A function f ∈ S is called uniformlystarlike of order α, α ∈ [0, 1) if

    Ref(z)− f(ζ)(z − ζ)f ′(z) ≥ α,(2.8)

    for any (z, ζ) from U × U . We denote by US∗(α) theclass of all this functions. We remark that US∗(0) =

    US∗.

    We also remark that the uniformly starlikeness of or-

    der α do not imply the starlikeness of order α. Indeed, if

    we consider ζ = 0 in the uniformly starlikeness of order

    α, with α ∈ (0, 1], it follow that Re f(z)zf ′(z) ≥ α, z ∈ U ,or equivalently, zf

    ′(z)f(z) take all values in the disc centered

    in 12α and with the radius12α . From the above do not

    follow that Rezf′(z)

    f(z) ≥ α, z ∈ U .

    48

  • Theorem 2.1.6 [32] Let f1(z) =z

    1−Az =

    z +∞∑

    n=2An−1zn, z ∈ U and α ∈ [0, 1).

    If |A| ≤ 1− α√2(α2 + 1)

    ,(2.9)

    then f1 ∈ US∗(α).

    Theorem 2.1.7 [32] Let f ∈ S, f(z) = z +∞∑

    n=2anz

    n

    and α ∈ [0, 1). If∞∑

    n=2

    √2(α2 + 1)

    1− α n|an| ≤ 1,(2.10)

    then f ∈ US∗(α).

    2.2 Uniformly convex functions

    The notion of uniformly convex function was introduced

    by A.W. Goodman in 1991 (see [23]) through analogy

    with the notion of uniformly starlike function.

    Definition 2.2.1 A function f is called uniformly con-

    vex in U if f ∈ Sc and for any circular arc γ from U ,with the center ζ also in U , the arc f(γ) is convex. We

    denote by UCV or USc the class of all this functions.

    49

  • For Γ(t) = f(γ) and γ = z(t), then f(γ) is convex if

    and only if

    Im

    [z′′(t)z′(t)

    +f ′′(z)f ′(z)

    z′(t)]≥ 0,(2.11)

    for any z onto Γ.

    We remark that for γ = Cr = {z ∈ C; |z| = r} weobtain the condition (1.19).

    If for the circular arc γ with the center ζ we consider

    z = ζ + reit, then z′(t) = i(z− ζ), z′′(t) = −(z− ζ), andfrom (2.11) we obtain:

    Theorem 2.2.1 [23] Let f ∈ S. Then f ∈ USc if andonly if

    1 + Re

    [f ′′(z)f ′(z)

    (z − ζ)]≥ 0,(2.12)

    for any (z, ζ) from U × U .

    Theorem 2.2.2 [23] The function

    f1(z) =z

    1− Az = z +∞∑

    n=2

    An−1zn, z ∈ U

    belong to the class USc if and only if |A| ≤ 1/3.

    50

  • Theorem 2.2.3 [23] The function

    f2(z) = z − Az2, z ∈ U

    belong to the class USc if and only if |A| ≤ 1/6.

    Theorem 2.2.4 [23] Let f ∈ S, f(z) = z +∞∑

    n=2anz

    n. If

    ∞∑n=2

    n(n− 1)|an| ≤ 13,(2.13)

    then f ∈ USc and the constant 1/3 can not be replacedwith a greater one.

    Theorem 2.2.5 [23] If f ∈ USc, f(z) = z +∞∑

    n=2

    anzn,

    then

    |an| ≤ 1n

    , n ≥ 2 .

    F. Ronning introduce in [46] the class SP which is

    important because it can be used to translate the results

    obtained from this class, directly to the class USc.

    Definition 2.2.2 SP = {F ∈ S∗|F (z) = zf ′(z),f ∈ USc}.

    51

  • Ma and Minda (see [31]) ,and independently, Ronning

    (see [46]) gives a characterization, which use only one

    variable, for the uniformly convex functions:

    Theorem 2.2.6 [31], [46] Let f ∈ S. Thenf ∈ USc if and only if:

    Re

    {1 +

    zf ′′(z)f ′(z)

    }>

    ∣∣∣∣zf ′′(z)f ′(z)

    ∣∣∣∣ , z ∈ U(2.14)

    For g(z) = zf ′(z) we obtain:

    Corollarly 2.2.1 [46] A function g ∈ S belong to theclass SP if and only if

    Rezg′(z)g(z)

    >

    ∣∣∣∣zg′(z)g(z)

    − 1∣∣∣∣ , z ∈ U.(2.15)

    From the geometric interpretation of the relation (2.15),

    we deduce that the class SP is the class of all the func-

    tion g ∈ S for which zg′(z)/g(z), z ∈ U take all thevalues into the parabolic domain

    Ω = {ω : |ω − 1| < Reω} =(2.16)

    {ω = u + iv; v2 < 2u− 1}.

    52

  • Theorem 2.2.7 [46] g(z) = z+anzn belong to the class

    SP if and only if

    |an| ≤ 12n− 1 .(2.17)

    Corollarly 2.2.2 f(z) = z + bnzn belong to the class

    USc if and only if

    |bn| ≤ 1n(2n− 1) .(2.18)

    Definition 2.2.3 [47] We will denote by SP (α, β),

    α > 0, β ∈ [0, 1) the class of all the functions f ∈ Swhich verify the condition:

    (2.19)∣∣∣∣zf ′(z)f(z)

    − (α + β)∣∣∣∣ ≤ Re

    zf ′(z)f(z)

    + α− β, z ∈ U.

    Geometric interpretation: f ∈ SP (α, β) if and only ifzf ′(z)/f(z), z ∈ U , take all the values into the parabolicdomain

    (2.20)

    Ωα,β = {ω : |ω − (α + β)| ≤ Reω + α− β} =53

  • {ω = u + iv : v2 ≤ 4α(u− β)}.

    Stankiewicz and Wisniowska introduce in [51], rela-

    tive to a hyperbolic domain, the following new class of

    functions:

    Definition 2.2.4 We say that the function f ∈ S belongto the class SH(α) if

    ∣∣∣∣zf ′(z)f(z)

    − 2α(√

    2− 1)∣∣∣∣ < Re

    {√2zf ′(z)f(z)

    }+

    2α(√

    2− 1)

    , z ∈ U , α > 0 .

    Remark 2.2.1 For the function f ∈ SH(α) the expres-sion

    zf ′(z)f(z)

    take all values into the interior of the posi-

    tive branch of the hyperbola v2 = 4αu + u2 , u > 0, and

    the function Hα, with Hα(0) = 1 and H′α(0) > 0, which

    is univalent and maps U into the above domain, is given

    by

    Hα(z) = (1 + 2α)

    √1 + bz

    1− bz − 2α

    where

    b = b(α) =1 + 4α− 4α2

    (1 + 2α)2.

    54

  • Definition 2.2.5 We say that a function f ∈ S is uni-formly convex of type α, α ≥ 0 if:

    Re

    {1 +

    zf ′′(z)f ′(z)

    }≥ α

    ∣∣∣∣zf ′′(z)f ′(z)

    ∣∣∣∣ , z ∈ U.(2.21)

    We denote by USc(α) (or k − UCV ) the class of allthis functions.

    Remark 2.2.2 The class USc(α) was introduced by Kanas

    and Wisniowska in [27] by using the following definition:

    Let 0 ≤ k < ∞. We say that a function f ∈ S isk-uniformly convex in U if the image of any circle arc

    γ contained in U , with the center ζ (|ζ| ≤ k), is convex.We denote by k − UCV the class of all this functions.

    We remark that USc(1) = USc and USc(0) = Sc.

    By this remark we obtain a continuously connection be-

    tween convexity and uniformly convexity.

    The geometric interpretation of the definition 2.2.5:

    f ∈ USc(α) if and only if 1 + zf ′′(z)/f ′(z) take all thevalues into the domain Dα, where Dα is:

    55

  • i) the elliptic region:(u− α2α2−1

    )2(

    αα2−1

    )2 +v2(1√

    α2−1

    )2 < 1, for α > 1

    ii) the parabolic region:

    v2 < 2u− 1, for α = 1

    iii) the hyperbolic region:(u + α

    2

    1−α2)2

    1−α2)2 −

    v2(1√

    1−α2)2 > 1, and u > 0,

    for 0 < α < 1

    iv) the half-plane u > 0, for α = 0.

    56

  • We also remark that USc(α) ⊂ Sc ( αα+1).

    Theorem 2.2.8 [27] Let f ∈ S, f(z) = z+∞∑

    j=2ajz

    j and

    α ≥ 0. If∞∑

    j=2

    j(j − 1)|aj| ≤ 1α + 2

    (2.22)

    then f ∈ USc(α). The constant 1/(α + 2) can not bereplaced be a greater one.

    Remark 2.2.3 Inspired by the class USc(α) Kanas and

    Wisniowska introduce, in [29], the class α−ST by usingthe following definition:

    α− ST := {f ∈ S : f(z) = zg′(z) , g ∈ USc(α)} ,

    α ≥ 0 , z ∈ U.

    In [53] the authors introduced the class of uniformly

    convex of order γ functions by using the following defi-

    nition:

    Definition 2.2.6 We say that a function f ∈ S is uni-formly convex of order γ ∈ [−1, 1) if

    (2.23)

    57

  • Re

    {1 +

    zf ′′(z)f ′(z)

    }≥

    ∣∣∣∣zf ′′(z)f ′(z)

    ∣∣∣∣ + γ, z ∈ U.

    We denote by USc[γ] the class of all this functions.

    The following subclasses are introduce by using the

    Sălăgean differential operator (see [50]):

    Dn : A → A , n ∈ N and D0f(z) = f(z)(2.24)

    D1f(z) = Df(z) = zf ′(z) , Dnf(z) = D(Dn−1f(z)

    )

    In 1999 I. Magdaş (see [33]), and independently, S.

    Kanas and T. Yaguchi (see [30]) introduce the class of

    n-uniformly starlike of type α functions:

    Definition 2.2.7 We say that a function f ∈ A is n-uniformly starlike of type α, α ≥ 0 and n ∈ N if:

    Re

    {Dn+1f(z)

    Dnf(z)

    }≥ α

    ∣∣∣∣Dn+1f(z)

    Dnf(z)− 1

    ∣∣∣∣ ,(2.25)

    for all z ∈ U .we denote by USn(α) the class of all this functions.

    We remark that US0(1) = SP, US1(1) = USc, US1(α) =

    USc(α), where USc(α) is the class defined by (2.21).

    58

  • Geometric interpretation of the relation (2.25): f ∈USn(α) if and only if D

    n+1f(z)/Dnf(z) take all values in

    the domain Dα, where Dα is a elliptic region for α > 1,

    a parabolic region for α = 1, a hyperbolic region for

    0 < α < 1, respectively the half-plan u > 0 for α = 0

    (see also the Definition (2.2.5)).

    From the above we remark that ReDn+1f(z)

    Dnf(z) >α

    α+1 .

    We have USn(α) ⊂ Sn(

    αα+1 , 1

    ) ⊂ S∗, and so we con-clude that the functions from USn(α) are univalent.

    Remark 2.2.4 In [30], S. Kanas and T. Yaguchi, the

    above mentioned functions are denominate (k, n)-uniformly

    convex functions and the class of all functions is denoted

    by (k, n)− UCV .In the same paper, the authors introduced also the

    class (k, n)− ST by the following definition:For f ∈ S, k ∈ [0,∞) and n ∈ N, we say that f

    belong to the class (k, n)− ST if

    Re

    (Dnf(z)

    f(z)

    )> k

    ∣∣∣∣Dnf(z)

    f(z)− 1

    ∣∣∣∣ , z ∈ U .

    59

  • I. Magdaş introduce in [34] the uniformly convex of

    type α and order γ functions and the n-uniformly star-

    like of order γ and type α functions:

    Definition 2.2.8 We say that a function f ∈ A is uni-formly convex of type α and order γ, α ≥ 0, γ ∈ [−1, 1),α + γ ≥ 0 if:

    Re

    {1 +

    zf ′′(z)f ′(z)

    }≥ α

    ∣∣∣∣zf ′′(z)f ′(z)

    ∣∣∣∣ + γ,(2.26)

    for all z ∈ U .

    We denote by USc(α, γ) the class of all this functions.

    We remark that USc(α, 0) = USc(α) and USc(1, γ) =

    USc[γ].

    Geometric interpretation of the relation (2.26):

    f ∈ USc(α, γ) if and only if 1 + zf ′′(z)f ′(z) take all values inthe domain Dα,γ, where Dα,γ is:

    i) a elliptic region:(u− α2−γα2−1

    )2[

    α(1−γ)α2−1

    ]2 +v2(

    1−γ√α2−1

    )2 < 1, for α > 1;

    60

  • ii) a parabolic region:

    v2 < 2(1− γ)u− (1− γ2), for α = 1;

    iii) a hyperbolic region:(u− γ−α21−α2

    )2[

    α(1−γ)1−α2

    ]2 +v2(

    1−γ√1−α2

    )2 > 1 and u > 0, for 0 < α < 1;

    iv) the half-plane u > γ, for α = 0

    We have Re{

    1 + zf′′(z)

    f ′(z)

    }> α+γα+1 .

    We also remark that USc(α, γ) ⊂ Sc (α+γα+1).

    Definition 2.2.9 We say that a function f ∈ A is n-uniformly starlike of order γ and type α, where

    61

  • α ≥ 0, γ ∈ [−1, 1), α + γ ≥ 0 and n ∈ N if

    ReDn+1f(z)

    Dnf(z)≥ α

    ∣∣∣∣Dn+1f(z)

    Dnf(z)− 1

    ∣∣∣∣ + γ,(2.27)

    for all z ∈ U .

    We denote by USn(α, γ) the class of all this functions.

    We remark that

    US1(α, γ) = USc(α, γ), US0(α, γ) = S

    ∗(γ),

    US0(1, γ) = SP

    (1− γ

    2,

    1 + γ

    2

    )and USn(α, 0) = USn(α).

    Geometric interpretation of the relation (2.27):

    f ∈ USn(α, γ) if and only if Dn+1f(z)/Dnf(z) take allvalues in the domain Dα,γ, where Dα,γ was defined at

    the geometric interpretation of the definition of the class

    USc(α, γ).

    We remember that for the functions f ∈ USn(α, γ)we have

    Re{Dn+1f(z)/Dnf(z)

    }> (α + γ)/(α + 1),

    and thus

    USn(α, γ) ⊂ Sn(

    α + γ

    1 + α

    )⊂ S∗.

    62

  • This mean that the functions from USn(α, γ) are univa-

    lent.

    Definition 2.2.10 Let f, g ∈ A; f(z) = z+∞∑

    j=2ajz

    j, z ∈

    U and g(z) = z+∞∑

    j=2bjz

    j, z ∈ U . We will denote by f ∗gthe convolution (or Hadamard) product of the functions

    f and g, defined by

    (f ∗ g)(z) = z +∞∑

    j=2

    ajbjzj, z ∈ U.(2.28)

    Definition 2.2.11 [48] We define the Ruscheweyh op-

    erator Rn : A → A, n ∈ N , z ∈ U , by:

    Rnf(z) =z

    (1− z)n+1 ∗ f(z) =z(zn−1f(z))(n)

    n!.(2.29)

    Remark 2.2.5 1. If f ∈ A, f(z) = z+∞∑

    j=2ajz

    j, z ∈ U ,then

    Rnf(z) = z +∞∑

    j=2

    Cnn+j−1ajzj, z ∈ U.(2.30)

    2. We remark that the inequality

    ReRn+1f(z)

    Rnf(z)>

    1

    2, z ∈ U(2.31)

    63

  • become for n = 1 the convexity condition.

    We will denote by Kn the class of all functions f ∈ Awhich satisfy (2.31).

    By using the Ruscheweyh operator, in [35], a new

    subclass of o uniformly convex functions is defined by:

    Definition 2.2.12 Let n ∈ N. We say that the functionf ∈ A belong to the class UKn(δ), δ ∈ [−1, 1), if:

    ReRn+1f(z)

    Rnf(z)≥

    ∣∣∣∣Rn+1f(z)

    Rnf(z)− 1

    ∣∣∣∣ + δ,(2.32)

    for all z ∈ U .

    Geometric interpretation: f ∈ UKn(δ), if and only ifRn+1f(z)/Rnf(z) take all values in the domain

    Ω 1−δ2 ,

    1+δ2

    not= Ωδ bounded by the parabola:

    v2 = 2(1− δ)u− (1− δ2).(2.33)

    The corresponding Carathéodory function is

    Qδ(z) = 1 +2(1− δ)

    π2

    (log

    1 +√

    z

    1−√z)2

    , z ∈ U.(2.34)

    64

  • We remark that the function Qδ is convex and satisfy

    ReQδ(z) >1 + δ

    2. We conclude that, f ∈ UKn(δ) if and

    only if Rn+1f(z)Rnf(z) ≺ Qδ(z).

    We remark that for n = 0 we have UK0(δ) =

    SP(1−δ

    2 ,1+δ2

    ), and for n = 1 and δ = 1/2, we have

    UK1(1/2) = USc.

    65

  • Chapter 3

    Subclasses of α-convex

    functions

    3.1 The subclasses UM(α) and UMα

    In [26] S. Kanas define the following subclass of α-uniformly

    convex functions:

    Definition 3.1.1 Let α ∈ [0, 1]. We say that a univa-lent function f is called α-uniformly convex if the image

    of every circle arc Γz contained in U and with the center

    ζ ∈ U , is a α-convex curve (see Definition 1.4.1) withrespect to f(ζ). We denote by UM(α) the class of all

    66

  • this functions. We remark that UM(α) ⊂ Mα, whereMα is the class of α-convex functions (see section 1.4)

    Theorem 3.1.1 [26] Let α ∈ [0, 1] and f be a univalentfunction. Then, f is a α-uniformly convex function if

    and only if

    Re

    {(1− α)(z − ζ)f

    ′(z)f(z)− f(ζ)

    (1 +

    (z − ζ)f ′′(z)f ′(z)

    )}> 0 , z, ζ ∈ U .

    Theorem 3.1.2 [26] If f is a α-uniformly convex func-

    tion and 0 ≤ β < α, then f is also a β-uniformly convexfunction, or briefly UM(α) ⊂ UM(β).

    In [36] I. Magdaş introduce the following subclass of

    α-uniformly convex functions:

    Definition 3.1.2 Let f ∈ A. We say that f is α-uniform convex function, α ∈ [0, 1] if

    Re

    {(1− α)zf

    ′(z)f(z)

    + α

    (1 +

    zf ′′(z)f ′(z)

    )}

    ≥∣∣∣∣(1− α)

    (zf ′(z)f(z)

    − 1)

    + αzf ′′(z)f ′(z)

    ∣∣∣∣ , z ∈ U.

    67

  • We denote this class with UMα.

    Remark 3.1.1 Geometric interpretation: f ∈ UMα ifand only if

    J(α, f ; z) = (1− α)zf′(z)

    f(z)+ α

    (1 +

    zf ′′(z)f ′(z)

    )

    takes all values in the parabolic region Ω = {w : |w−1| ≤Re w} = {w = u + iv ; v2 ≤ 2u − 1}. We have UM0 =SP (see definition 2.2.2). Also, we have UMα ⊂ Mα,where Mα is the class of α-convex functions.

    3.2 The subclass UDn,α(β, γ)

    The results included in this section are obtained in [1].

    Definition 3.2.1 Let α ∈ [0, 1] and n ∈ N. We saythat f ∈ A is in the class UDn,α(β, γ) , β ≥ 0 , γ ∈[−1, 1) , β + γ ≥ 0, if

    Re

    [(1− α)D

    n+1f(z)

    Dnf(z)+ α

    Dn+2f(z)

    Dn+1f(z)

    ]

    ≥ β∣∣∣∣(1− α)

    Dn+1f(z)

    Dnf(z)+ α

    Dn+2f(z)

    Dn+1f(z)− 1

    ∣∣∣∣ + γ ,

    68

  • where Dn is the differential operator defined by (2.24).

    Remark 3.2.1 We have UDn,0(β, γ) = USn(β, γ) ⊂ S∗

    ,UD0,α(1, 0) = UMα and UD0,1(β, γ) = USc(β, γ) ⊂

    Sc(

    β + γ

    β + 1

    ), where USn(β, γ) is the class given in the

    definition 2.2.9, UMα is the class of α-uniformly convex

    functions defined in the previously section, USc(β, γ) is

    the class of the uniformly convex functions of type β and

    order γ (see definition 2.2.8) and Sc(δ) is the class of

    convex functions of order δ (see definition1.3.5).

    Remark 3.2.2 Geometric interpretation:

    f ∈ UDn,α(β, γ) if and only if

    Jn(α, f ; z) = (1− α)Dn+1f(z)

    Dnf(z)+ α

    Dn+2f(z)

    Dn+1f(z)

    takes all values in the convex domain Dβ,γ, where Dβ,γ is

    defined at the geometric interpretation of the definition

    2.2.8.

    Theorem 3.2.1 For all α, α′ ∈ [0, 1] with α < α′, wehave UDn,α′(β, γ) ⊂ UDn,α(β, γ).

    69

  • Proof. From f ∈ UDn,α′(β, γ) we have

    Re

    [(1− α′)D

    n+1f(z)

    Dnf(z)+ α′

    Dn+2f(z)

    Dn+1f(z)

    ]

    ≥ β∣∣∣∣(1− α′)

    Dn+1f(z)

    Dnf(z)+ α′

    Dn+2f(z)

    Dn+1f(z)− 1

    ∣∣∣∣ + γ .

    With the notationsDn+1f(z)

    Dnf(z)= p(z), where p(z) =

    1 + p1z + · · ·, we have

    zp′(z) = z

    (Dn+1f(z)

    )′ ·Dnf(z)−Dn+1f(z) · (Dnf(z))′(Dnf(z))2

    =Dn+2f(z)

    Dnf(z)−

    (Dn+1f(z)

    Dnf(z)

    )2,

    zp′(z)p(z)

    =Dn+2f(z)

    Dn+1f(z)− D

    n+1f(z)

    Dnf(z),

    and thus we obtain

    Jn(α′, f ; z) = p(z) + α′ · zp

    ′(z)p(z)

    .

    Now we have that p(z) + α′ · zp′(z)

    p(z)takes all values in

    the convex domain Dβ,γ which is included in right half

    plane.

    If we consider h ∈ Hu(U), with h(0) = 1, which mapsthe unit disc U into the convex domain Dβ,γ, we have

    70

  • Reh(z) > 0 and from hypothesis α′ > 0. From here

    follows that Re1

    α′· h(z) > 0. In this conditions from

    Theorem 1.6.1 , with δ = 0 we obtain p(z) ≺ h(z), orp(z) take all values in Dβ,γ.

    If we consider the function g : [0, α′] → C,g(u) = p(z) + u · zp

    ′(z)p(z)

    , with g(0) = p(z) ∈ Dβ,γ andg(α′) ∈ Dβ,γ. Since the geometric image of g(α) is on thesegment obtained by the union of the geometric image

    of g(0) and g(α′), we have g(α) ∈ Dβ,γ, or

    p(z) + α · zp′(z)

    p(z)∈ Dβ,γ .

    Thus Jn(α, f ; z) takes all values in Dβ,γ, or f ∈ UDn,α(β, γ).

    Remark 3.2.3 From Theorem 3.2.1 we have UDn,α(β, γ)

    ⊂ UDn,0(β, γ) for all α ∈ [0, 1], and from Remark 3.2.1we obtain that the functions from the class UDn,α(β, γ)

    are univalent.

    71

  • Let consider the integral operator La : A → A definedas (see [44]):

    f(z) = LaF (z) =1 + a

    za

    z∫

    0

    F (t)·ta−1dt, a ∈ C, Re a ≥ 0.

    (3.1)

    Remark 3.2.4 If we take a = 1, 2, 3, ... in the above

    definition, we obtain the Bernardi integral operator (see

    [15]).

    Theorem 3.2.2 If F (z) ∈ UDn,α(β, γ) then f(z) =La(F )(z) ∈ USn(β, γ), where La is the integral opera-tor defined by (3.1) and the class USn(β, γ) is given in

    the definition 2.2.9.

    Proof. From (3.1) we have

    (1 + a)F (z) = af(z) + zf ′(z)

    By means of the application of the linear operator Dn+1

    we obtain

    (1 + a)Dn+1F (z) = aDn+1f(z) + Dn+1(zf ′(z))

    72

  • or

    (1 + a)Dn+1F (z) = aDn+1f(z) + Dn+2f(z) .

    Thus:

    Dn+1F (z)

    DnF (z)=

    Dn+2f(z) + aDn+1f(z)

    Dn+1f(z) + aDnf(z)

    =

    Dn+2f(z)

    Dn+1f(z)· D

    n+1f(z)

    Dnf(z)+ a · D

    n+1f(z)

    Dnf(z)

    Dn+1f(z)

    Dnf(z)+ a

    .

    With the notationDn+1f(z)

    Dnf(z)= p(z) where p(z) = 1+

    p1z + ..., we have:

    zp′(z) = z ·(

    Dn+1f(z)

    Dnf(z)

    )′

    =z(Dn+1f(z)

    )′ ·Dnf(z)−Dn+1f(z) · z (Dnf(z))′(Dnf(z))2

    =Dn+2f(z) ·Dnf(z)− (Dn+1f(z))2

    (Dnf(z))2

    and

    1

    p(z)· zp′(z) = D

    n+2f(z)

    Dn+1f(z)− D

    n+1f(z)

    Dnf(z)=

    Dn+2f(z)

    Dn+1f(z)− p(z) .

    73

  • It follows:

    Dn+2f(z)

    Dn+1f(z)= p(z) +

    1

    p(z)· zp′(z) .

    Thus we obtain:

    Dn+1F (z)

    DnF (z)=

    p(z) ·(zp′(z) · 1p(z) + p(z)

    )+ a · p(z)

    p(z) + a

    = p(z) +1

    p(z) + a· zp′(z) .

    If we denoteDn+1F (z)

    DnF (z)= q(z), with q(0) = 1, and

    we consider h ∈ Hu(U), with h(0) = 1, which maps theunit disc U into the convex domain Dβ,γ, we have from

    F (z) ∈ UDn,α(β, γ) (see Remark 3.2.2):

    q(z) + α · zq′(z)

    q(z)≺ h(z) .

    From Theorem 1.6.1 , with δ = 0 we obtain q(z) ≺ h(z),or

    p(z) +1

    p(z) + a· zp′(z) ≺ h(z) .

    Using the hypothesis and the construction of the func-

    tion h(z) we obtain from Theorem 1.6.1 p(z) ≺ h(z) or74

  • f(z) ∈ USn(β, γ) (see the geometric interpretation ofthe definition 2.2.9).

    Remark 3.2.5 From Theorem 3.2.2 with α = 0 we ob-

    tain the Theorem 3.1 from [7] which assert that the in-

    tegral operator La, defined by (3.1), preserve the class

    USn(β, γ).

    3.3 The subclasses UMα(q) and UDn,α(q)

    In the beginning of this section we will recall some re-

    sults due to D. Blezu (see [18]):

    Definition 3.3.1 The function f ∈ A is n-starlike withrespect to convex domain included in right half plane D if

    the differential expressionDn+1f(z)

    Dnf(z)takes values in the

    domain D, where Dn is the differential operator defined

    by (2.24).

    If we consider q(z) an univalent function with

    q(0) = 1, Re q(z) > 0, q′(0) > 0 which maps the unit

    75

  • disc U into the convex domain D we have:

    Dn+1f(z)

    Dnf(z)≺ q(z).

    We note by S∗n(q) the set of all these functions.

    The following results are obtained in [2].

    Let q(z) be an univalent function with q(0) = 1,

    q′(0) > 0, which maps the unit disc U into a convex

    domain included in right half plane D.

    Definition 3.3.2 Let f ∈ A and α ∈ [0, 1]. We say thatf is α-uniform convex function with respect to D, if

    J(α, f ; z) = (1− α)zf′(z)

    f(z)+ α

    (1 +

    zf ′′(z)f ′(z)

    )≺ q(z).

    We denote this class with UMα(q).

    Remark 3.3.1 Geometric interpretation: f ∈ UMα(q)if and only if J(α, f ; z) take all values in the convex

    domain included in right half plan D.

    Remark 3.3.2 We have UMα(q) ⊂ Mα, where Mα isthe well know class of α-convex function. If we take

    D = Ω (see Remark 3.1.1) we obtain the class UMα.

    76

  • Remark 3.3.3 From the above definition it easily re-

    sults that q1(z) ≺ q2(z) implies UMα(q1) ⊂ UMα(q2).

    Theorem 3.3.1 For all α, α′ ∈ [0, 1] with α < α′ wehave UMα′(q) ⊂ UMα(q).

    Proof. From f ∈ UMα′(q) we have

    J(α′, f ; z) = (1− α)zf′(z)

    f(z)+ α

    (1 +

    zf ′′(z)f ′(z)

    )≺ q(z),

    (3.2)

    where q(z) is univalent in U with q(0) = 1, q′(0) > 0, and

    maps the unit disc U into the convex domain included

    in right half plane D.

    With notationzf ′(z)f(z)

    = p(z), where p(z) = 1+p1z+...

    we have:

    J(α′, f ; z) = p(z) + α′ · zp′(z)

    p(z).

    From (3.2) we have p(z) + α′ · zp′(z)

    p(z)≺ q(z) with

    p(0) = q(0) and Re q(z) > 0, z ∈ U .In this conditions from Theorem 1.6.1, with δ = 0, we

    obtain p(z) ≺ q(z), or p(z) take all values in D.77

  • If we consider the function g : [0, α′] → C,g(u) = p(z) + u · zp

    ′(z)p(z)

    , with g(0) = p(z) ∈ D andg(α′) = J(α′, f ; z) ∈ D. Since the geometric imageof g(α) is on the segment obtained by the union of the

    geometric image of g(0) and g(α′), we have g(α) ∈ D orp(z) + α

    zp′(z)p(z)

    ∈ D.Thus J(α, f ; z) take all values in D, or

    J(α, f ; z) ≺ q(z). This means f ∈ UMα(q).

    Theorem 3.3.2 If F (z) ∈ UMα(q) thenf(z) = La(F )(z) ∈ S∗0(q), where La is the integral oper-ator defined by (3.1) and α ∈ [0, 1].

    Proof. From (3.1) we have

    (1 + a)F (z) = af(z) + zf ′(z).

    With notationzf ′(z)f(z)

    = p(z), where p(z) = 1+p1z+...

    we have

    zF ′(z)F (z)

    = p(z) +zp′(z)

    p(z) + a.

    78

  • If we denotezF ′(z)F (z)

    = h(z), with h(0) = 1, we have

    from F (z) ∈ UMα(q) (see Definition 3.3.2):

    h(z) + α · zh′(z)

    h(z)≺ q(z),

    where q(z) is univalent un U with q(0) = 1, q′(z) > 0 and

    maps the unit disc U into the convex domain included

    in right half plane D.

    From Theorem 1.6.1 we obtain h(z) ≺ q(z) orp(z) +

    zp′(z)p(z) + a

    ≺ q(z).Using the hypothesis and the construction of the func-

    tion q(z) we obtain from Theorem 1.6.1zf ′(z)f(z)

    = p(z) ≺ q(z) or f(z) ∈ S∗0(q) ⊂ S∗.

    Definition 3.3.3 Let f ∈ A, α ∈ [0, 1] and n ∈ N.We say that f is α − n-uniformly convex function withrespect to D if

    Jn(α, f ; z) = (1− α)Dn+1f(z)

    Dnf(z)+ α

    Dn+2f(z)

    Dn+1f(z)≺ q(z),

    where Dn is the differential operator defined by (2.24).

    We denote this class with UDn,α(q).

    79

  • Remark 3.3.4 Geometric interpretation: f ∈ UDn,α(q)if and only if Jn(α, f ; z) take all values in the convex do-

    main included in right half plane D.

    Remark 3.3.5 We have UD0,α(q) = UMα(q) and if in

    the above definition we consider D = Dβ,γ (see Remark

    3.2.2) we obtain the class UDn,α(β, γ).

    Remark 3.3.6 It is easy to see that q1(z) ≺ q2(z) im-plies UDn,α(q1) ⊂ UDn,α(q2).

    Theorem 3.3.3 For all α, α′ ∈ [0, 1] with α < α′ wehave UDn,α′(q) ⊂ UDn,α(q).

    Proof. From f ∈ UDn,α′(q) we have:

    Jn(α′, f ; z) = (1−α)D

    n+1f(z)

    Dnf(z)+ α

    Dn+2f(z)

    Dn+1f(z)≺ q(z),

    (3.3)

    where q(z) is univalent in U with q(0) = 1, q′(0) > 0, and

    maps the unit disc U into the convex domain included

    in right half plane D.

    80

  • With notationDn+1f(z)

    Dnf(z)= p(z), where

    p(z) = 1 + p1z + ... we have

    Jn(α′, f ; z) = p(z) + α′ · zp

    ′(z)p(z)

    .

    From (3.3) we have p(z) + α′ · zp′(z)

    p(z)≺ q(z) with

    p(0) = q(0) and Re q(z) > 0, z ∈ U . In this conditionfrom Theorem 1.6.1 we obtain p(z) ≺ q(z), or p(z) takeall values in D.

    If we consider the function g : [0, α′] → C,g(u) = p(z) + u · zp

    ′(z)p(z)

    , with g(0) = p(z) ∈ D andg(α′) = Jn(α′, f ; z) ∈ D, it easy to see thatg(α) = p(z) + α

    zp′(z)p(z)

    ∈ D.Thus we have Jn(α, f ; z) ≺ q(z) or f ∈ UDn,α(q).

    Theorem 3.3.4 If F (z) ∈ UDn,α(q) thenf(z) = La(F )(z) ∈ S∗n(q), where La is the integral oper-ator defined by (3.1).

    Proof. From (3.1) we have

    (1 + a)F (z) = af(z) + zf ′(z). By means of the applica-

    81

  • tion of the linear operator Dn+1 we obtain:

    (1 + a)Dn+1F (z) = aDn+1f(z) + Dn+1(zf ′(z))

    or

    (1 + a)Dn+1F (z) = aDn+1f(z) + Dn+2f(z).

    With notationDn+1f(z)

    Dnf(z)= p(z), where

    p(z) = 1 + p1z + ..., we have:

    Dn+1F (z)

    DnF (z)= p(z) +

    1

    p(z) + a· zp′(z).

    If we denoteDn+1F (z)

    DnF (z)= h(z), with h(0) = 1, we

    have from F ∈ UDn,α(q):

    h(z) + αzh′(z)h(x)

    ≺ q(z),

    where q(z) is univalent in U with q(0) = 1, q′(0) > 0, and

    maps the unit disc U into the convex domain included

    in right half plane D.

    From Theorem 1.6.1 we obtain h(z) ≺ q(z) orp(z) +

    zp′(z)p(z) + a

    ≺ q(z).Using the hypothesis we obtain from Theorem 1.6.1

    p(z) ≺ q(z) or f(z) ∈ S∗n(q).82

  • Remark 3.3.7 If we consider D = Dβ,γ in Theorem

    3.3.3 and Theorem 3.3.4 we obtain the main results from

    the previously section and if we take D = Dβ,γ and α = 0

    in Theorem 3.3.4 we obtain the Theorem 3.1 from [7].

    3.4 The subclass Mλ,α(q)

    For the main results from this section we need to recall

    here the following definitions and theorems:

    Definition 3.4.1 [3] Let λ ∈ R , λ ≥ 0 andf(z) = z +

    ∞∑

    j=2

    ajzj. We define the generalized Sălăgean

    operator by Dλ : A → A

    Dλf(z) = z +∞∑

    j=2

    jλajzj .

    Remark 3.4.1 [3] It is easy to observe that the general-

    ized Sălăgean operator defined above is a linear operator.

    Also, we observe that for λ ∈ N we obtain the Sălăgeandifferential operator.

    83

  • Definition 3.4.2 [3] Let q(z) ∈ Hu(U), with q(0) = 1and q(U) = D, where D is a convex domain contained in

    the right half plane. We say that a function f(z) ∈ A isa λ-q-starlike function if

    Dλ+1f(z)

    Dλf(z)≺ q(z). We denote

    this class by S∗λ(q).

    Definition 3.4.3 [8] Let q(z) ∈ Hu(U), with q(0) = 1and q(U) = D, where D is a convex domain contained

    in the right half plane. We say that a function f(z) ∈ Ais a λ-q-convex function if

    Dλ+2f(z)

    Dλ+1f(z)≺ q(z). We denote

    this class by Scλ(q).

    The main results of this section are obtained in [4].

    Definition 3.4.4 Let α ∈ [0, 1], q(z) ∈ Hu(U), withq(0) = 1 and q(U) = D, where D is a convex domain

    contained in the right half plane. We say that a function

    f(z) ∈ A is a λ-q-α-convex function if

    Jλ(α, f ; z) = (1− α)Dλ+1f(z)

    Dλf(z)+ α

    Dλ+2f(z)

    Dλ+1f(z)≺ q(z) .

    We denote this class with Mλ,α(q).

    84

  • Remark 3.4.2 Geometric interpretation: f(z) ∈ Mλ,α(q)if and only if Jλ(α, f ; z) take all values in the convex do-

    main D contained in the right half-plane.

    Remark 3.4.3 It is easy to observe that if we choose

    different function q(z) we obtain variously classes of

    α-convex functions, such as (for example), for λ = 0,

    the class of α-convex functions, the class of α-uniform

    convex functions with respect to a convex domain (see

    the previously section), and, for λ = n ∈ N, the classUDn,α(β, γ), β ≥ 0, γ ∈ [−1, 1), β + γ ≥ 0 (see the sec-tion 3.2), the class of α-n-uniformly convex functions

    with respect to a convex domain (see the previously sec-

    tion).

    Remark 3.4.4 We have Mλ,0(q) = S∗λ(q) and

    Mλ,1(q) = Scλ(q).

    Remark 3.4.5 For q1(z) ≺ q2(z) we haveMλ,α(q1) ⊂ Mλ,α(q2) .From the above we obtain Mλ,α(q) ⊂ Mλ,α

    (1 + z

    1− z)

    .

    85

  • Theorem 3.4.1 Let λ ∈ R , λ ≥ 0.For all α, α′ ∈ [0, 1], with α < α′ we haveMλ,α′(q) ⊂ Mλ,α(q) .

    Proof. From f(z) ∈ Mλ,α′(q) we have

    Jλ(α′, f ; z) = (1− α′)D

    λ+1f(z)

    Dλf(z)+ α′

    Dλ+2f(z)

    Dλ+1f(z)≺ q(z) ,

    (3.4)

    where q(z) is univalent in U with q(0) = 1 and maps the

    unit disc U into the convex domain D contained in the

    right half-plane.

    With notation

    p(z) =Dλ+1f(z)

    Dλf(z),

    where

    p(z) = 1 + p1z + . . . ,

    we have

    p(z) + α′ · zp′(z)

    p(z)

    =Dλ+1f(z)

    Dλf(z)+ α′

    Dλf(z)

    Dλ+1f(z)

    86

  • ·z(Dλ+1f(z)

    )′Dλf(z)−Dλ+1f(z) (Dλf(z))′

    (Dλf(z))2

    =Dλ+1f(z)

    Dλf(z)+ α′

    Dλf(z)

    Dλ+1f(z)

    (z(Dλ+1f(z)

    )′Dλf(z)

    −Dλ+1f(z)

    Dλf(z)· z

    (Dλf(z)

    )′Dλf(z)

    )

    =Dλ+1f(z)

    Dλf(z)+ α′ · D

    λf(z)

    Dλ+1f(z)

    z

    (z +

    ∞∑j=2

    jλ+1ajzj

    )′

    Dλf(z)

    −Dλ+1f(z)

    Dλf(z)·z

    (z +

    ∞∑

    j=2

    jλajzj

    )′

    Dλf(z)

    =Dλ+1f(z)

    Dλf(z)+ α′ · D

    λf(z)

    Dλ+1f(z)

    z

    (1 +

    ∞∑j=2

    j(jλ+1aj)zj−1

    )

    Dλf(z)

    87

  • −Dλ+1f(z)

    Dλf(z)·z

    (1 +

    ∞∑j=2

    j(jλaj)zj−1

    )

    Dλf(z)

    =Dλ+1f(z)

    Dλf(z)+ α′ · D

    λf(z)

    Dλ+1f(z)

    z +∞∑

    j=2

    jλ+2ajzj

    Dλf(z)

    −Dλ+1f(z)

    Dλf(z)·z +

    ∞∑j=2

    jλ+1ajzj

    Dλf(z)

    =Dλ+1f(z)

    Dλf(z)+ α′ · D

    λf(z)

    Dλ+1f(z)

    (Dλ+2f(z)

    Dλf(z)− D

    λ+1f(z)

    Dλf(z)

    ·Dλ+1f(z)

    Dλf(z)

    )

    =Dλ+1f(z)

    Dλf(z)+ α′ · D

    λ+2f(z)

    Dλ+1f(z)− α′ · D

    λ+1f(z)

    Dλf(z)

    =Dλ+1f(z)

    Dλf(z)(1− α′) + α′ · D

    λ+2f(z)

    Dλ+1f(z)= Jλ(α

    ′, f ; z)

    88

  • From (3.4) we have

    p(z) +zp′(z)1

    α′· p(z)

    ≺ q(z) ,

    with p(0) = q(0), Re q(z) > 0 , z ∈ U , and α′ > 0. Inthis conditions from Theorem 1.6.1 we obtain

    p(z) ≺ q(z) or p(z) take all values in D.If we consider the function g : [0, α′] → C,

    g(u) = p(z) + u · zp′(z)

    p(z),

    with g(0) = p(z) ∈ D and g(α′) = Jλ(α′, f ; z) ∈ D, iteasy to see that

    g(α) = p(z) + α · zp′(z)

    p(z)∈ D , 0 ≤ α < α′ .

    Thus we have

    Jλ(α, f ; z) ≺ q(z)

    or

    f(z) ∈ Mλ,α(q) .

    Remark 3.4.6 From the above theorem we have, for ev-

    ery α ∈ [0, 1], that Mλ,α(q) ⊂ S∗λ(q).89

  • Remark 3.4.7 If we consider λ = 0 we obtain the The-

    orem 3.3.1 from the section 3.3. Also, for λ = n ∈ N,we obtain the Theorem 3.3.3 from the previously section.

    Remark 3.4.8 If we consider D = Dβ,γ (see the geo-

    metric interpretation of the definition 2.2.8) in the above

    theorem we obtain the Theorem 3.2.1 from the section

    3.2.

    Theorem 3.4.2 If F (z) ∈ Mλ,α(q) thenf(z) = LaF (z) ∈ S∗λ(q), where La is the integral operatordefined by (3.1).

    Proof. From (3.1) we have

    (1 + a)F (z) = af(z) + zf ′(z)

    and, by using the linear operator Dλ+1, we obtain

    (1 + a)Dλ+1F (z) = aDλ+1f(z) + Dλ+1 (zf ′(z))

    = aDλ+1f(z) + Dλ+1

    (z +

    ∞∑

    j=2

    jajzj

    )

    = aDλ+1f(z) + z +∞∑

    j=2

    jλ+1(jaj)zj

    90

  • = aDλ+1f(z) + Dλ+2f(z)

    or

    (1 + a)Dλ+1F (z) = aDλ+1f(z) + Dλ+2f(z) .

    Similarly, we obtain

    (1 + a)DλF (z) = aDλf(z) + Dλ+1f(z) .

    Then

    Dλ+1F (z)

    DλF (z)=

    Dλ+2f(z)

    Dλ+1f(z)· D

    λ+1f(z)

    Dλf(z)+ a · D

    λ+1f(z)

    Dλf(z)

    Dλ+1f(z)

    Dλf(z)+ a

    .

    With notation

    Dλ+1f(z)

    Dλf(z)= p(z) , p(0) = 1 ,

    we obtain

    Dλ+1F (z)

    DλF (z)=

    Dλ+2f(z)

    Dλ+1f(z)· p(z) + a · p(z)

    p(z) + a(3.5)

    We have

    Dλ+2f(z)

    Dλ+1f(z)=

    Dλ+2f(z)

    Dλf(z)· D

    λf(z)

    Dλ+1f(z)=

    1

    p(z)· D

    λ+2f(z)

    Dλf(z)(3.6)

    91

  • Also, we have

    Dλ+2f(z)

    Dλf(z)=

    z +∞∑

    j=2

    jλ+2ajzj

    z +∞∑

    j=2

    jλajzj

    and

    zp′(z) =z(Dλ+1f(z)

    )′Dλf(z)

    − Dλ+1f(z)

    Dλf(z)· z

    (Dλf(z)

    )′Dλf(z)

    =

    =

    z +∞∑

    j=2

    jλ+2ajzj

    z +∞∑

    j=2

    jλajzj

    − p(z) ·z +

    ∞∑j=2

    jλ+1ajzj

    z +∞∑

    j=2

    jλajzj

    =

    =Dλ+2f(z)

    Dλf(z)− p(z) · D

    λ+1f(z)

    Dλf(z).

    Thus

    zp′(z) =Dλ+2f(z)

    Dλf(z)− p(z)2

    orDλ+2f(z)

    Dλf(z)= p(z)2 + zp′(z) .

    From (3.6) we obtain

    Dλ+2f(z)

    Dλ+1f(z)=

    1

    p(z)

    [p(z)2 + zp′(z)

    ]= p(z) +

    zp′(z)p(z)

    .

    92

  • From (3.5) we obtain

    Dλ+1F (z)

    DλF (z)=

    (p(z) +

    zp′(z)p(z)

    )· p(z) + a · p(z)

    p(z) + a

    = p(z) +zp′(z)

    p(z) + a

    If we denote

    Dλ+1F (z)

    DλF (z)= h(z) , with h(0) = 1 ,

    we have from F (z) ∈ Mλ,α(q) (see the proof of the abovetheorem):

    Jλ(α, F ; z) = h(z) + α · zh′(z)

    h(z)≺ q(z) .

    Using the hypothesis we obtain from Theorem 1.6.1

    h(z) ≺ q(z)

    or

    p(z) +zp′(z)

    p(z) + a≺ q(z) .

    By using the Theorem 1.6.1 and the hypothesis we have

    p(z) ≺ q(z)93

  • orDλ+1f(z)

    Dλf(z)≺ q(z) .

    This means that f(z) ∈ S∗λ(q) .

    Remark 3.4.9 If we consider λ = 0 we obtain the The-

    orem 3.3.2 from previously section. Also, for λ = n ∈ N,we obtain the Theorem 3.3.4 from the section 3.3.

    Remark 3.4.10 If we consider D = Dβ,γ (see remark

    3.4.8) in the above theorem we obtain the Theorem 3.2.2

    from the section 3.2.

    3.5 The subclass MLn,α(q)

    In the first part of this section we will introduce some

    usefully definitions and remarks:

    Definition 3.5.1 [14] Let n ∈ N and λ ≥ 0. We denotewith Dnλ the operator defined by

    Dnλ : A → A ,

    D0λf(z) = f(z) , D1λf(z) = (1−λ)f(z)+λzf ′(z) = Dλf(z) ,

    94

  • Dnλf(z) = Dλ(Dn−1λ f(z)

    ).

    Remark 3.5.1 [14] We observe that Dnλ is a linear op-

    erator and for f(z) = z +∞∑

    j=2

    ajzj we have

    Dnλf(z) = z +∞∑

    j=2

    (1 + (j − 1)λ)n ajzj .

    Also, it is easy to observe that if we consider λ = 1 in

    the above definition we obtain the Sălăgean differential

    operator (see (2.24)).

    Definition 3.5.2 [9] Let q(z) ∈ Hu(U), withq(0) = 1 and q(U) = D, where D is a convex domain

    contained in the right half plane, n ∈ N and λ ≥ 0. Wesay that a function f(z) ∈ A is in the class SL∗n(q) ifDn+1λ f(z)

    Dnλf(z)≺ q(z) , z ∈ U .

    Remark 3.5.2 Geometric interpretation: f(z) ∈ SL∗n(q)if and only if

    Dn+1λ f(z)

    Dnλf(z)take all values in the convex do-

    main D contained in the right half-plane.

    95

  • Definition 3.5.3 [10] Let q(z) ∈ Hu(U), with q(0) =1 and q(U) = D, where D is a convex domain con-

    tained in the right half plane, n ∈ N and λ ≥ 0. Wesay that a function f(z) ∈ A is in the class SLcn(q) ifDn+2λ f(z)

    Dn+1λ f(z)≺ q(z) , z ∈ U .

    Remark 3.5.3 Geometric interpretation: f(z) ∈ SLcn(q)if and only if

    Dn+2λ f(z)

    Dn+1λ f(z)take all values in the convex do-

    main D contained in the right half-plane.

    The main results of this section are obtained in [5].

    Definition 3.5.4 Let q(z) ∈ Hu(U), with q(0) = 1,q(U) = D, where D is a convex domain contained in

    the right half plane, n ∈ N, λ ≥ 0 and α ∈ [0, 1]. Wesay that a function f(z) ∈ A is in the class MLn,α(q) if

    Jn,λ(α, f ; z) = (1− α)Dn+1λ f(z)

    Dnλf(z)+ α

    Dn+2λ f(z)

    Dn+1λ f(z)≺ q(z)

    , z ∈ U .

    Remark 3.5.4 Geometric interpretation:

    f(z) ∈ MLn,α(q) if and only if Jn,λ(α, f : z) take all96

  • values in the convex domain D contained in the right

    half-plane.

    Remark 3.5.5 It is easy to observe that if we choose

    different function q(z) we obtain variously classes of α-

    convex functions, such as (for example), for λ = 1 and

    n = 0, the class of α-convex functions, the class of α-

    uniform convex functions with respect to a convex do-

    main (see the section 3.3), and, for λ = 1, the class

    UDn,α(β, γ), β ≥ 0, γ ∈ [−1, 1), β + γ ≥ 0 (see the sec-tion 3.2), the class of α-n-uniformly convex functions

    with respect to a convex domain (see the section 3.3).

    Remark 3.5.6 We have MLn,0(q) = SL∗n(q) and

    MLn,1(q) = SLcn(q).

    Remark 3.5.7 For q1(z) ≺ q2(z) we haveMLn,α(q1) ⊂ MLn,α(q2) . From the above we obtainMLn,α(q) ⊂ MLn,α

    (1 + z

    1− z)

    .

    Theorem 3.5.1 For all α, α′ ∈ [0, 1], with α < α′, wehave MLn,α′(q) ⊂ MLn,α(q) .

    97

  • Proof. From f(z) ∈ MLn,α′(q) we have

    Jn,λ(α′, f ; z) = (1− α′)D

    n+1λ f(z)

    Dnλf(z)+ α′

    Dn+2λ f(z)

    Dn+1λ f(z)≺ q(z) ,

    (3.7)

    where q(z) is univalent in U with q(0) = 1 and maps the

    unit disc U into the convex domain D contained in the

    right half-plane.

    With notation

    p(z) =Dn+1λ f(z)

    Dnλf(z),

    where

    p(z) = 1 + p1z + . . . and f(z) = z +∞∑

    j=2

    ajzj

    we have

    p(z) + α′ · λ · zp′(z)

    p(z)

    =Dn+1λ f(z)

    Dnλf(z)+ α′λ

    Dnλf(z)

    Dn+1λ f(z)

    ·z(Dn+1λ f(z)

    )′Dnλf(z)−Dn+1λ f(z) (Dnλf(z))′

    (Dnλf(z))2

    =Dn+1λ f(z)

    Dnλf(z)+ α′λ

    Dnλf(z)

    Dn+1λ f(z)

    (z

    (Dn+1λ f(z)

    )′Dnλf(z)

    98

  • −Dn+1λ f(z)

    Dnλf(z)· z (D

    nλf(z))

    Dnλf(z)

    )

    =Dn+1λ f(z)

    Dnλf(z)+ α′λ

    · Dnλf(z)

    Dn+1λ f(z)

    z

    (z +

    ∞∑j=2

    (1 + (j − 1)λ)n+1 ajzj)′

    Dnλf(z)

    −Dn+1λ f(z)

    Dnλf(z)·z

    (z +

    ∞∑

    j=2

    (1 + (j − 1)λ)n ajzj)′

    Dnλf(z)

    =Dn+1λ f(z)

    Dnλf(z)+ α′λ

    · Dnλf(z)

    Dn+1λ f(z)

    z

    (1 +

    ∞∑j=2

    j (1 + (j − 1)λ)n+1 ajzj−1)

    Dnλf(z)

    99

  • −Dn+1λ f(z)

    Dnλf(z)·z

    (1 +

    ∞∑j=2

    j (1 + (j − 1)λ)n ajzj−1)

    Dnλf(z)

    or

    p(z) + α′ · λ · zp′(z)

    p(z)=

    Dn+1λ f(z)

    Dnλf(z)+ α′λ(3.8)

    · Dnλf(z)

    Dn+1λ f(z)

    z +∞∑

    j=2

    j (1 + (j − 1)λ)n+1 ajzj

    Dnλf(z)

    −Dn+1λ f(z)

    Dnλf(z)·z +

    ∞∑j=2

    j (1 + (j − 1)λ)n ajzj

    Dnλf(z)

    We have

    z +∞∑

    j=2

    j (1 + (j − 1)λ)n+1 ajzj

    = z +∞∑

    j=2

    ((j − 1) + 1) (1 + (j − 1)λ)n+1 ajzj

    100

  • = z +∞∑

    j=2

    (1 + (j − 1)λ)n+1 ajzj

    +∞∑

    j=2

    (j − 1) (1 + (j − 1)λ)n+1 ajzj

    = z + Dn+1λ f(z)− z +∞∑

    j=2

    (j − 1) (1 + (j − 1)λ)n+1 ajzj

    = Dn+1λ f(z) +1

    λ

    ∞∑j=2

    ((j − 1)λ) (1 + (j − 1)λ)n+1 ajzj

    = Dn+1λ f(z)

    +1

    λ

    ∞∑j=2

    (1 + (j − 1)λ− 1) (1 + (j − 1)λ)n+1 ajzj

    = Dn+1λ f(z)−1

    λ

    ∞∑j=2

    (1 + (j − 1)λ)n+1 ajzj

    +1

    λ

    ∞∑j=2

    (1 + (j − 1)λ)n+2 ajzj

    = Dn+1λ f(z)−1

    λ

    (Dn+1λ f(z)− z

    )

    +1

    λ

    (Dn+2λ f(z)− z

    )

    = Dn+1λ f(z)−1

    λDn+1λ f(z) +

    z

    λ+

    1

    λDn+2λ f(z)−

    z

    λ

    101

  • =λ− 1

    λDn+1λ f(z) +

    1

    λDn+2λ f(z)

    =1

    λ

    ((λ− 1)Dn+1λ f(z) + Dn+2λ f(z)

    ).

    Similarly we have

    z +∞∑

    j=2

    j (1 + (j − 1)λ)n ajzj

    =1

    λ

    ((λ− 1)Dnλf(z) + Dn+1λ f(z)

    ).

    From (3.8) we obtain

    p(z) + α′ · λ · zp′(z)

    p(z)

    =Dn+1λ f(z)

    Dnλf(z)+ α′λ

    Dnλf(z)

    Dn+1λ f(z)

    1

    λ·(

    (λ− 1)Dn+1λ f(z)

    Dnλf(z)

    +Dn+2λ f(z)

    Dnλf(z)− D

    n+1λ f(z)

    Dnλf(z)(λ− 1)−

    (Dn+1λ f(z)

    Dnλf(z)

    )2)

    =Dn+1λ f(z)

    Dnλf(z)+ α′

    Dn+2λ f(z)

    Dn+1λ f(z)− α′D

    n+1λ f(z)

    Dnλf(z)

    =Dn+1λ f(z)

    Dnλf(z)(1− α′) + α′D

    n+2λ f(z)

    Dn+1λ f(z)= Jn,λ(α

    ′, f ; z)

    From (3.7) we have

    p(z) +zp′(z)1

    α′λ· p(z)

    ≺ q(z) ,

    102

  • with p(0) = q(0), Re q(z) > 0 , z ∈ U , α′ > 0 andλ ≥ 0. In this conditions from Theorem 1.6.1 we obtainp(z) ≺ q(z) or p(z) take all values in D.

    If we consider the function g : [0, α′] → C,

    g(u) = p(z) + u · λzp′(z)

    p(z),

    with g(0) = p(z) ∈ D and g(α′) = Jn,λ(α′, f ; z) ∈ D, iteasy to see that

    g(α) = p(z) + α · λzp′(z)

    p(z)∈ D , 0 ≤ α < α′ .

    Thus we have

    Jn,λ(α, f ; z) ≺ q(z)

    or

    f(z) ∈ MLn,α(q) .

    From the above theorem we have

    Corollarly 3.5.1 For every n ∈ N and α ∈ [0, 1], we103

  • have

    MLn,α(q) ⊂ MLn,0(q) = SL∗n(q) .

    Remark 3.5.8 If we consider λ = 1 and n = 0 we

    obtain the Theorem 3.3.1 from the section 3.3. Also, for

    λ = 1 and n ∈ N, we obtain the Theorem 3.3.3 from thesame section.

    Remark 3.5.9 If we consider λ = 1 and D = Dβ,γ (see

    the geometric interpretation of the definition 2.2.8) in

    the above theorem we obtain the Theorem 3.2.1 from the

    section 3.2.

    Theorem 3.5.2 Let n ∈ N, α ∈ [0, 1] and λ ≥ 1 . IfF (z) ∈ MLn,α(q) then f(z) = LaF (z) ∈ SL∗n(q), whereLa is the integral operator defined by (2.24).

    Proof. From (2.24) we have

    (1 + a)F (z) = af(z) + zf ′(z)

    104

  • and, by using the linear operator Dn+1λ and if we consider

    f(z) =∞∑

    j=2

    ajzj, we obtain

    (1 + a)Dn+1λ F (z) = aDn+1λ f(z) + D

    n+1λ

    (z +

    ∞∑

    j=2

    jajzj

    )

    = aDn+1λ f(z) + z +∞∑

    j=2

    (1 + (j − 1)λ)n+1 jajzj

    We have (see the proof of the above theorem)

    z +∞∑

    j=2

    j (1 + (j − 1)λ)n+1 ajzj(3.9)

    =1

    λ

    ((λ− 1)Dn+1λ f(z) + Dn+2λ f(z)

    )

    Thus

    (1 + a)Dn+1λ F (z) = aDn+1λ f(z)

    +1

    λ

    ((λ− 1)Dn+1λ f(z) + Dn+2λ f(z)

    )

    =

    (a +

    λ− 1λ

    )Dn+1λ f(z) +

    1

    λDn+2λ f(z)

    or

    λ(1+a)Dn+1λ F (z) = ((a + 1)λ− 1) Dn+1λ f(z)+Dn+2λ f(z) .105

  • Similarly, we obtain

    λ(1 + a)DnλF (z) = ((a + 1)λ− 1) Dnλf(z) + Dn+1λ f(z) .

    ThenDn+1λ F (z)

    DnλF (z)

    =

    Dn+2λ f(z)

    Dn+1λ f(z)· D

    n+1λ f(z)

    Dnλf(z)+ ((a + 1)λ− 1) · D

    n+1λ f(z)

    Dnλf(z)

    Dn+1λ f(z)

    Dnλf(z)+ ((a + 1)λ− 1)

    .

    With notation

    Dn+1λ f(z)

    Dnλf(z)= p(z) , p(0) = 1 ,

    we obtainDn+1λ F (z)

    DnλF (z)(3.10)

    =

    Dn+2λ f(z)

    Dn+1λ f(z)· p(z) + ((a + 1)λ− 1) · p(z)

    p(z) + ((a + 1)λ− 1)Also, we obtain

    Dn+2λ f(z)

    Dn+1λ f(z)=

    Dn+2λ f(z)

    Dnλf(z)· D

    nλf(z)

    Dn+1λ f(z)=

    1

    p(z)· D

    n+2λ f(z)

    Dnλf(z)(3.11)

    106

  • We have

    Dn+2λ f(z)

    Dnλf(z)=

    z +∞∑

    j=2

    (1 + (j − 1)λ)n+2 ajzj

    z +∞∑

    j=2

    (1 + (j − 1)λ)n ajzj

    and

    zp′(z) =z(Dn+1λ f(z)

    )′Dnλf(z)

    − Dn+1λ f(z)

    Dnλf(z)· z (D

    nλf(z))

    Dnλf(z)

    =

    z

    (1 +

    ∞∑j=2

    (1 + (j − 1)λ)n+1 jajzj−1)

    Dnλf(z)

    −p(z) ·z

    (1 +

    ∞∑

    j=2

    (1 + (j − 1)λ)n jajzj−1)

    Dnλf(z)

    or

    zp′(z) =

    z +∞∑

    j=2

    j (1 + (j − 1)λ)n+1 ajzj

    Dnλf(z)(3.12)

    −p(z) ·z +

    ∞∑

    j=2

    j (1 + (j − 1)λ)n ajzj

    Dnλf(z).

    107

  • By using (3.9) and (3.12) we obtain

    zp′(z) =1

    λ

    ((λ− 1)Dn+1λ f(z) + Dn+2λ f(z)

    Dnλf(z)

    −p(z)(λ− 1)Dnλf(z) + D

    n+1λ f(z)

    Dnλf(z)

    )

    =1

    λ

    ((λ− 1)p(z) + D

    n+2λ f(z)

    Dnλf(z)− p(z) ((λ− 1) + p(z))

    )

    =1

    λ

    (Dn+2λ f(z)

    Dnλf(z)− p(z)2

    )

    Thus

    λzp′(z) =Dn+2λ f(z)

    Dnλf(z)− p(z)2

    orDn+2λ f(z)

    Dnλf(z)= p(z)2 + λzp′(z) .

    From (3.11) we obtain

    Dn+2λ f(z)

    Dn+1λ f(z)=

    1

    p(z)

    (p(z)2 + λzp′(z)

    ).

    Then, from (3.10), we obtain

    Dn+1λ F (z)

    DnλF (z)=

    p(z)2 + λzp′(z) + ((a + 1)λ− 1) p(z)p(z) + ((a + 1)λ− 1)

    = p(z) + λzp′(z)

    p(z) + ((a + 1)λ− 1) ,

    108

  • where a ∈ C, Rea ≥ 0 and λ ≥ 1 .If we denote

    Dn+1λ F (z)

    DnλF (z)= h(z), with h(0) = 1, we

    have from F (z) ∈ MLn,α(q) (see the proof of the aboveTheorem):

    Jn,λ(α, F ; z) = h(z) + α · λ · zh′(z)

    h(z)≺ q(z)

    Using the hypothesis, from Theorem 1.6.1, we obtain

    h(z) ≺ q(z)

    or

    p(z) + λzp′(z)

    p(z) + ((a + 1)λ− 1) ≺ q(z) .

    By using the Theorem 1.6.1 and the hypothesis we

    have

    p(z) ≺ q(z)

    orDn+1λ f(z)

    Dnλf(z)≺ q(z) .

    This means f(z) = LaF (z) ∈ SL∗n(q) .

    Remark 3.5.10 If we consider λ = 1 and n = 0 we

    obtain the Theorem 3.3.2 from the section 3.3. Also, for

    109

  • λ = 1 and n ∈ N, we obtain the Theorem 3.3.4 from thesame section.

    Remark 3.5.11 If we consider λ = 1 and D = Dβ,γ

    (see remark 3.4.6) in the above theorem we obtain the

    Theorem 3.2.2 from the section 3.2.

    3.6 The subclass MLβ,α(q)

    For the main results of this section we will need the

    following definitions and theorems:

    Definition 3.6.1 [11] Let β, λ ∈ R, β ≥ 0, λ ≥ 0 andf(z) = z+

    ∞∑j=2

    ajzj. We denote by Dβλ the linear operator

    defined by

    Dβλ : A → A ,

    Dβλf(z) = z +∞∑

    j=2

    (1 + (j − 1)λ)β ajzj .

    Definition 3.6.2 [11] Let q(z) ∈ Hu(U), with q(0) = 1and q(U) = D, where D is a convex domain contained

    in the right half plane, β, λ ∈ R, β ≥ 0 and λ ≥ 0. We110

  • say that a function f(z) ∈ A is in the class SL∗β(q) if

    Dβ+1λ f(z)

    Dβλf(z)≺ q(z) , z ∈ U .

    Theorem 3.6.1 [11] Let β, λ ∈ R, β ≥ 0 and λ ≥ 1 . IfF (z) ∈ SL∗β(q) then f(z) = LaF (z) ∈ SL∗β(q), where Lai