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

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

(α

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)

+α

(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