γλώσσες
Σελίδες
Νομικός
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) the
class 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) ⊂ E
then g f ∈ Hu(D).
1.1.2) The Koebe function f(z) = z(1−z)2 , z ∈ U
is univalent in U .
Theorem 1.1.1 [25] If f ∈ Hu(D), then f ′(z) 6= 0 for
all 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 that
this 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 series
expansion (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 +α1
ζ+
α2
ζ2 + ... +αn
ζn+ ..., |ζ| > 1.
A function ϕ from∑
will verify the normalization
conditions ϕ(∞) = ∞ and ϕ′(∞) = 1. We will also
denote 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 +a3
z+ ... ∈
∑
and g(z) 6= 0 for all z ∈ U−, because f ∈ S has no poles.
Conversely, if g ∈ ∑,
g(z) = z + b0 +b1
z+ ...
and c ∈ C∞\g(U−), then the function
f(z) =1
g(1
z
)− c
=z
1 + (b0 − c)z + ...= z + (c− b0)z
2 + ... ∈ 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 if
g(z) = z + b0 + e2iθ/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 by
w0 = −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∈S
f(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− r
1 + 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∈S
f(U(0, r))
U(0, r2) =⋃
f∈S
f(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+a2z2+ ..., z ∈ U , then |an| ≤ n, n ≥ 2 . The extremal
functions 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 the
circle 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 the
above condition will hold also for |z| ≤ r. From the
above, we conclude that from f starlike function onto
the circle z ∈ C : |z| = r, we obtain that f will
be starlike onto every circle z ∈ C : |z| = r′, where
0 < r′ < r.
Definition 1.2.2 We define the radii of starlikeness for
the function f by
r∗(f) = sup
r; Re
zf ′(z)
f(z)> 0, |z| ≤ r
.(1.11)
If r∗(f) ≥ 1 we will say that f is a starlike function onto
the 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 to
a constant.
17
2) The condition Rezf ′(z)
f(z)> 0, z ∈ U , do not assure
that the function f will be univalent in the unit disk.
If we impose the additional condition f ′(0) 6= 0, then
the condition Rezf ′(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 ifzf ′(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− r
1 + r≤
∣∣∣∣zf ′(z)
f(z)
∣∣∣∣ ≤1 + r
1− r(1.15)
where z ∈ U, |z| = r, and the extremal function is the
Koebe 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 exist
a 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), denoted
by 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 starlike
of 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 called
strongly starlike of order α, 0 < α ≤ 1 if the following
inequality hold∣∣∣∣arg
zf ′(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 all
0 < |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 convex
in 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 all
0 < |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 Strohhacker)[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. The
equality 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 the
Koebe constant for the class Sc is 1/2.
Definition 1.3.5 We say that the function f ∈ A is
convex of order α, 0 ≤ α, < 1, if the following inequality
hold
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) = supr; ReJ(α, f ; z) > 0, |z| ≤ r.27
If rα(f) ≥ 1 we say that the function f is α-convex in
the 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 + z
1− 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 conditionf(z)f ′(z)
z6= 0, z ∈ U will
also hold.
Theorem 1.4.1 [43] For α, β ∈ R with
0 ≤ β/α ≤ 1, we have Mα ⊂ Mβ.
Corollarly 1.4.1 For all α ∈ [0, 1], we have
Sc ⊂ Mα ⊂ S∗.
Theorem 1.4.2 [43] If α > 0, then f ∈ Mα if and only
if 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 only
if 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-convex
of 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−√−α
2 +√−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, with
p(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 we
want 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 function
ψ, 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 function
ψ, 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 from
U 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, for
every p which satisfy (1.39).
Definition 1.5.4 A dominant q which satisfy q ≺ q for
every 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 origin
and with the radii r0, U r0=
z ∈ C; |z| ≤ r0.
Lemma 1.5.1 Let z0 ∈ U , r0 = |z0| and f(z) = anzn+
+an+1zn+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 +
pnzn + ... be a analytic function in U , with p(z) 6≡ a and
n ≥ 1. If there exist z0 ∈ U and ζ0 ∈ ∂U\E(q) such that
p(z0) = q(ζ0) and p(Ur0) ⊂ q(U), where r0 = |z0|, then
there 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 + pnzn + ... be a analytic function in U with p(z) 6≡ a
and n ≥ 1. If p 6≺ q then there exist z0 = r0eiθ0 ∈ U and
ζ0 ∈ ∂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 following
admissibility 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 be
a 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 ⊂ Ω, then ψn(Ω, 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+pnzn+... 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 function
in 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) = a
and ψ(a, 0, 0; 0) = = h(0). If p(z) = a + pnzn + ... 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, with
ψ(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) and
hρ(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 differential
equation
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 number
ρ ∈ (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 analytic
function 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 a
analytic 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 satisfy
the 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 Strohacker 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 assume
that 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 best
dominant.
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 subordination
p(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 a
circle 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 = 0
it is easy to see that we obtain the condition (1.10).
For z = ζ + reit we have z′(t) = i(z − ζ), and we
obtain:
Theorem 2.1.1 [22] Let f ∈ S. Then f ∈ US∗ if and
only 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) it
is 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 uniformly
starlike 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) the
class 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 radius 1
2α . From the above do not
follow that Rezf ′(z)f(z) ≥ α, z ∈ U .
48
Theorem 2.1.6 [32] Let f1(z) = z1−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− αn|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 we
obtain 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), and
from (2.11) we obtain:
Theorem 2.2.1 [23] Let f ∈ S. Then f ∈ USc if and
only 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| ≤ 1
3,(2.13)
then f ∈ USc and the constant 1/3 can not be replaced
with a greater one.
Theorem 2.2.5 [23] If f ∈ USc, f(z) = z +∞∑
n=2
anzn,
then
|an| ≤ 1
n, 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. Then
f ∈ 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 the
class 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 the
values 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| ≤ 1
2n− 1.(2.17)
Corollarly 2.2.2 f(z) = z + bnzn belong to the class
USc if and only if
|bn| ≤ 1
n(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 ∈ S
which verify the condition:
(2.19)∣∣∣∣zf ′(z)
f(z)− (α + β)
∣∣∣∣ ≤ Rezf ′(z)
f(z)+ α− β, z ∈ U.
Geometric interpretation: f ∈ SP (α, β) if and only if
zf ′(z)/f(z), z ∈ U , take all the values into the parabolic
domain
(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 belong
to 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-
sionzf ′(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 all
this 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 is
k-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 the
values into the domain Dα, where Dα is:
55
i) the elliptic region:(u− α2
α2−1
)2
(α
α2−1
)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 + α2
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| ≤ 1
α + 2(2.22)
then f ∈ USc(α). The constant 1/(α + 2) can not be
replaced 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 using
the 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
Salagean 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. Magdas (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 Dn+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
(α
α+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. Magdas 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 in
the domain Dα,γ, where Dα,γ is:
i) a elliptic region:(u− α2−γ
α2−1
)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−α2
1−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+γ
α+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 all
values 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
ReDn+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−1ajz
j, 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 ∈ A
which 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 function
f ∈ 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 if
Rn+1f(z)/Rnf(z) take all values in the domain
Ω 1−δ2 , 1+δ
2
not= Ωδ bounded by the parabola:
v2 = 2(1− δ)u− (1− δ2).(2.33)
The corresponding Caratheodory 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) with
respect to f(ζ). We denote by UM(α) the class of all
66
this functions. We remark that UM(α) ⊂ Mα, where
Mα is the class of α-convex functions (see section 1.4)
Theorem 3.1.1 [26] Let α ∈ [0, 1] and f be a univalent
function. 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 convex
function, or briefly UM(α) ⊂ UM(β).
In [36] I. Magdas 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
)+ α
zf ′′(z)
f ′(z)
∣∣∣∣ , z ∈ U.
67
We denote this class with UMα.
Remark 3.1.1 Geometric interpretation: f ∈ UMα if
and 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 say
that f ∈ A is in the class UDn,α(β, γ) , β ≥ 0 , γ ∈[−1, 1) , β + γ ≥ 0, if
Re
[(1− α)
Dn+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′, we
have UDn,α′(β, γ) ⊂ UDn,α(β, γ).
69
Proof. From f ∈ UDn,α′(β, γ) we have
Re
[(1− α′)
Dn+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)− Dn+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 maps
the 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), or
p(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 and
g(α′) ∈ Dβ,γ. Since the geometric image of g(α) is on the
segment 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 3.2.1
we obtain that the functions from the class UDn,α(β, γ)
are univalent.
71
Let consider the integral operator La : A → A defined
as (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) =
Dn+2f(z)
Dn+1f(z)− Dn+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) · 1
p(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 the
unit 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) or
74
f(z) ∈ USn(β, γ) (see the geometric interpretation of
the 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 with
respect 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 that
f 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 is
the 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′ we
have 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 and
g(α′) = J(α′, f ; z) ∈ D. Since the geometric image
of 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 or
p(z) + αzp′(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) then
f(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) or
p(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 with
respect 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′ we
have UDn,α′(q) ⊂ UDn,α(q).
Proof. From f ∈ UDn,α′(q) we have:
Jn(α′, f ; z) = (1−α)
Dn+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 condition
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′) = Jn(α′, f ; z) ∈ D, it easy to see that
g(α) = p(z) + αzp′(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) then
f(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) or
p(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 and
f(z) = z +∞∑
j=2
ajzj. We define the generalized Salagean
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 Salagean operator defined above is a linear operator.
Also, we observe that for λ ∈ N we obtain the Salagean
differential operator.
83
Definition 3.4.2 [3] Let q(z) ∈ Hu(U), with q(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-starlike function ifDλ+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) = 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-convex function ifDλ+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), with
q(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 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 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 have
Mλ,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 have
Mλ,α′(q) ⊂ Mλ,α(q) .
Proof. From f(z) ∈ Mλ,α′(q) we have
Jλ(α′, f ; z) = (1− α′)
Dλ+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′ · Dλf(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′ · Dλf(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′ · Dλf(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′ · Dλf(z)
Dλ+1f(z)
(Dλ+2f(z)
Dλf(z)− Dλ+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. In
this 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, it
easy 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) then
f(z) = LaF (z) ∈ S∗λ(q), where La is the integral operator
defined 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)· Dλf(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 above
theorem):
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 denote
with Dnλ the operator defined by
Dnλ : A → A ,
D0λf(z) = f(z) , D1
λf(z) = (1−λ)f(z)+λzf ′(z) = Dλf(z) ,
94
Dnλf(z) = Dλ
(Dn−1
λ 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 Salagean differential
operator (see (2.24)).
Definition 3.5.2 [9] Let q(z) ∈ Hu(U), with
q(0) = 1 and q(U) = D, where D is a convex domain
contained in the right half plane, n ∈ N and λ ≥ 0. We
say that a function f(z) ∈ A is in the class SL∗n(q) ifDn+1
λ f(z)
Dnλf(z)
≺ q(z) , z ∈ U .
Remark 3.5.2 Geometric interpretation: f(z) ∈ SL∗n(q)
if and only ifDn+1
λ 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. We
say that a function f(z) ∈ A is in the class SLcn(q) if
Dn+2λ f(z)
Dn+1λ f(z)
≺ q(z) , z ∈ U .
Remark 3.5.3 Geometric interpretation: f(z) ∈ SLcn(q)
if and only ifDn+2
λ 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]. We
say that a function f(z) ∈ A is in the class MLn,α(q) if
Jn,λ(α, f ; z) = (1− α)Dn+1
λ f(z)
Dnλf(z)
+ αDn+2
λ 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 all
96
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 have
MLn,α(q1) ⊂ MLn,α(q2) . From the above we obtain
MLn,α(q) ⊂ MLn,α
(1 + z
1− z
).
Theorem 3.5.1 For all α, α′ ∈ [0, 1], with α < α′, we
have MLn,α′(q) ⊂ MLn,α(q) .
97
Proof. From f(z) ∈ MLn,α′(q) we have
Jn,λ(α′, f ; z) = (1− α′)
Dn+1λ f(z)
Dnλf(z)
+ α′Dn+2
λ 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+1
λ 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+1
λ f(z)
Dnλf(z)
+ α′λDn
λf(z)
Dn+1λ f(z)
·z(Dn+1
λ f(z))′
Dnλf(z)−Dn+1
λ f(z) (Dnλf(z))′
(Dnλf(z))2
=Dn+1
λ f(z)
Dnλf(z)
+ α′λDn
λf(z)
Dn+1λ f(z)
(z
(Dn+1
λ f(z))′
Dnλf(z)
98
−Dn+1λ f(z)
Dnλf(z)
· z (Dnλf(z))′
Dnλf(z)
)
=Dn+1
λ 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+1
λ 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+1
λ f(z)− z)
+1
λ
(Dn+2
λ f(z)− z)
= Dn+1λ f(z)− 1
λDn+1
λ f(z) +z
λ+
1
λDn+2
λ f(z)− z
λ
101
=λ− 1
λDn+1
λ f(z) +1
λDn+2
λ f(z)
=1
λ
((λ− 1)Dn+1
λ f(z) + Dn+2λ f(z)
).
Similarly we have
z +∞∑
j=2
j (1 + (j − 1)λ)n ajzj
=1
λ
((λ− 1)Dn
λf(z) + Dn+1λ f(z)
).
From (3.8) we obtain
p(z) + α′ · λ · zp′(z)
p(z)
=Dn+1
λ f(z)
Dnλf(z)
+ α′λDn
λf(z)
Dn+1λ f(z)
1
λ·(
(λ− 1)Dn+1
λ f(z)
Dnλf(z)
+Dn+2
λ f(z)
Dnλf(z)
− Dn+1λ f(z)
Dnλf(z)
(λ− 1)−(
Dn+1λ f(z)
Dnλf(z)
)2)
=Dn+1
λ f(z)
Dnλf(z)
+ α′Dn+2
λ f(z)
Dn+1λ f(z)
− α′Dn+1
λ f(z)
Dnλf(z)
=Dn+1
λ f(z)
Dnλf(z)
(1− α′) + α′Dn+2
λ 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 and
λ ≥ 0. In this 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 · λzp′(z)
p(z),
with g(0) = p(z) ∈ D and g(α′) = Jn,λ(α′, f ; z) ∈ D, it
easy 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], we
103
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 the
same 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 . If
F (z) ∈ MLn,α(q) then f(z) = LaF (z) ∈ SL∗n(q), where
La 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+1
λ f(z) + Dn+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+1
λ f(z) + Dn+2λ f(z)
)
Thus
(1 + a)Dn+1λ F (z) = aDn+1
λ f(z)
+1
λ
((λ− 1)Dn+1
λ f(z) + Dn+2λ f(z)
)
=
(a +
λ− 1
λ
)Dn+1
λ f(z) +1
λDn+2
λ f(z)
or
λ(1+a)Dn+1λ F (z) = ((a + 1)λ− 1) Dn+1
λ f(z)+Dn+2λ f(z) .
105
Similarly, we obtain
λ(1 + a)DnλF (z) = ((a + 1)λ− 1) Dn
λf(z) + Dn+1λ f(z) .
ThenDn+1
λ F (z)
DnλF (z)
=
Dn+2λ f(z)
Dn+1λ f(z)
· Dn+1λ f(z)
Dnλf(z)
+ ((a + 1)λ− 1) · Dn+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+1
λ 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+2
λ f(z)
Dnλf(z)
· Dnλf(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+1
λ f(z))′
Dnλf(z)
− Dn+1λ f(z)
Dnλf(z)
· z (Dnλf(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+1
λ f(z) + Dn+2λ f(z)
Dnλf(z)
−p(z)(λ− 1)Dn
λf(z) + Dn+1λ f(z)
Dnλf(z)
)
=1
λ
((λ− 1)p(z) +
Dn+2λ f(z)
Dnλf(z)
− p(z) ((λ− 1) + p(z))
)
=1
λ
(Dn+2
λ f(z)
Dnλf(z)
− p(z)2)
Thus
λzp′(z) =Dn+2
λ f(z)
Dnλf(z)
− p(z)2
orDn+2
λ 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 denoteDn+1
λ F (z)
DnλF (z)
= h(z), with h(0) = 1, we
have from F (z) ∈ MLn,α(q) (see the proof of the above
Theorem):
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+1
λ 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 the
same 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 and
f(z) = z+∞∑
j=2
ajzj. We denote by Dβ
λ 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) = 1
and q(U) = D, where D is a convex domain contained
in the right half plane, β, λ ∈ R, β ≥ 0 and λ ≥ 0. We
110
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 . If
F (z) ∈ SL∗β(q) then f(z) = LaF (z) ∈ SL∗β(q), where La
is the integral operator defined by (3.1).
Definition 3.6.3 [12] Let q(z) ∈ Hu(U), with q(0) = 1
and q(U) = D, where D is a convex domain contained
in the right half plane, β, λ ∈ R, β ≥ 0 and λ ≥ 0. We
say that a function f(z) ∈ A is in the class SLcβ(q) if
Dβ+2λ f(z)
Dβ+1λ f(z)
≺ q(z) , z ∈ U .
Theorem 3.6.2 [12] Let β, λ ∈ R, β ≥ 0 and λ ≥ 1 . If
F (z) ∈ SLcβ(q) then f(z) = LaF (z) ∈ SLc
β(q), where La
is the integral operator defined by (3.1).
The main results of this section are obtained in [6].
Definition 3.6.4 Let q(z) ∈ Hu(U), with q(0) = 1,
q(U) = D, where D is a convex domain contained in
111
the right half plane, β ≥ 0, λ ≥ 0 and α ∈ [0, 1]. We
say that a function f(z) ∈ A is in the class MLβ,α(q) if
Jβ,λ(α, f ; z) = (1− α)Dβ+1
λ f(z)
Dβλf(z)
+ αDβ+2
λ f(z)
Dβ+1λ f(z)
≺ q(z) ,
z ∈ U .
Remark 3.6.1 Geometric interpretation:
f(z) ∈ MLβ,α(q) if and only if Jβ,λ(α, f : z) take all
values in the convex domain D contained in the right
half-plane.
Remark 3.6.2 We have MLβ,0(q) = SL∗β(q) and
MLβ,1(q) = SLcβ(q).
Remark 3.6.3 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 = 1
and β = 0, the class of α-convex functions, the class
of α-uniform convex functions with respect to a con-
vex domain (see the section 3.3), and, for λ = 1 and
β = n ∈ N, the class UDn,α(b, γ), b ≥ 0, γ ∈ [−1, 1),
112
b+γ ≥ 0 (see the section 3.2), the class of α-n-uniformly
convex functions with respect to a convex domain (see
the section 3.3).
Remark 3.6.4 For q1(z) ≺ q2(z) we have
MLβ,α(q1) ⊂ MLβ,α(q2) . From the above we obtain
MLβ,α(q) ⊂ MLβ,α
(1 + z
1− z
).
Remark 3.6.5 It is easy to observe that for every
β ≥ 0, α ∈ [0, 1] and λ ≥ 0 we have id(z) ∈ MLβ,α(q),
where id(z) = z, z ∈ U .
Theorem 3.6.3 Let q(z) ∈ Hu(U), with q(0) = 1,
q(U) = D, where D is a convex domain contained in the
right half plane, β ≥ 0 and λ ≥ 0. For all α, α′ ∈ [0, 1],
with α < α′, we have MLβ,α′(q) ⊂ MLβ,α(q) .
Proof. From f(z) ∈ MLβ,α′(q) we have
Jβ,λ(α′, f ; z) = (1− α′)
Dβ+1λ f(z)
Dβλf(z)
+ α′Dβ+2
λ f(z)
Dβ+1λ f(z)
≺ q(z) ,
(3.13)
113
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.
Define the function
p(z) =Dβ+1
λ f(z)
Dβλf(z)
= 1 + p1z + · · ·
for f(z) ∈ A with
f(z) = z +∞∑
j=2
ajzj.
Note that
z(Dβ+1
λ f(z))′ = z +
∞∑
j=2
j (1 + (j − 1)λ)β+1 ajzj
= z +∞∑
j=2
((j − 1) + 1) (1 + (j − 1)λ)β+1 ajzj
= z +∞∑
j=2
(1 + (j − 1)λ)β+1 ajzj
+∞∑
j=2
(j − 1) (1 + (j − 1)λ)β+1 ajzj
= z + Dβ+1λ f(z)− z +
∞∑j=2
(j − 1) (1 + (j − 1)λ)β+1 ajzj
114
= Dβ+1λ f(z) +
1
λ
∞∑j=2
((j − 1)λ) (1 + (j − 1)λ)β+1 ajzj
= Dβ+1λ f(z)
+1
λ
∞∑j=2
(1 + (j − 1)λ− 1) (1 + (j − 1)λ)β+1 ajzj
= Dβ+1λ f(z)− 1
λ
∞∑j=2
(1 + (j − 1)λ)β+1 ajzj
+1
λ
∞∑j=2
(1 + (j − 1)λ)β+2 ajzj
= Dβ+1λ f(z)− 1
λ
(Dβ+1
λ f(z)− z)
+1
λ
(Dβ+2
λ f(z)− z)
= Dβ+1λ f(z)− 1
λDβ+1
λ f(z) +z
λ+
1
λDβ+2
λ f(z)− z
λ
=λ− 1
λDβ+1
λ f(z) +1
λDβ+2
λ f(z)
=1
λ
((λ− 1)Dβ+1
λ f(z) + Dβ+2λ f(z)
).
Similarly we have
z(Dβ
λf(z))′ = z +
∞∑j=2
j (1 + (j − 1)λ)β ajzj
=1
λ
((λ− 1)Dβ
λf(z) + Dβ+1λ f(z)
).
115
Thus we see that
p(z) + α′λzp′(z)
p(z)=
Dβλf(z)
Dβ+1λ f(z)
+α′λ
((λ− 1)Dβ+1
λ f(z) + Dβ+2λ f(z)
λDβ+1λ f(z)
−(λ− 1)Dβλf(z) + Dβ+1
λ f(z)
λDβλf(z)
)
= (1− α′)Dβ+1
λ f(z)
Dβλf(z)
+ α′Dβ+2
λ f(z)
Dβ+1λ f(z)
= Jβ,λ(α′, f ; z).
From (3.13) we have
p(z) +zp′(z)1
α′λ· p(z)
≺ q(z) ,
with p(0) = q(0), Re q(z) > 0 , z ∈ U , α′ > 0 and
λ ≥ 0. In this 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 · λzp′(z)
p(z),
116
with g(0) = p(z) ∈ D and g(α′) = Jβ,λ(α′, f ; z) ∈ D, it
easy to see that
g(α) = p(z) + α · λzp′(z)
p(z)∈ D , 0 ≤ α < α′ .
Thus we have
Jβ,λ(α, f ; z) ≺ q(z)
or
f(z) ∈ MLβ,α(q) .
From the above theorem we have
Corollarly 3.6.1 For every β ≥ 0, λ ≥ 0 and
α ∈ [0, 1], we have
MLβ,α(q) ⊂ MLβ,0(q) = SL∗β(q)
.
Theorem 3.6.4 Let q(z) ∈ Hu(U), with q(0) = 1,
q(U) = D, where D is a convex domain contained in
117
the right half plane, β ≥ 0, α ∈ [0, 1] and λ ≥ 1 . If
F (z) ∈ MLβ,α(q) then f(z) = LaF (z) ∈ SL∗β(q), where
La is the integral operator defined by (3.1).
Proof. From (3.1) we have
(1 + a)F (z) = af(z) + zf ′(z) .
Note that
(1 + a)Dβ+1λ F (z) = aDβ+1
λ f(z) + z(Dβ+1
λ f(z))′
= aDβ+1λ f(z) +
1
λ
((λ− 1)Dβ+1
λ f(z) + Dβ+2λ f(z)
)
or
λ(1+a)Dβ+1λ F (z) = ((a + 1)λ− 1) Dβ+1
λ f(z)+Dβ+2λ f(z)
and
λ(1 + a)DβλF (z) = ((a + 1)λ− 1) Dβ
λf(z) + Dβ+1λ f(z).
With the following definition for p(z),
Dβ+1λ f(z)
Dβλf(z)
= p(z) , p(0) = 1 ,
118
we obtain
zp′(z)Dβλf(z) + zp(z)
(Dβ
λf(z))′ = z
(Dβ+1
λ f(z))′.
This implies that
λzp(z)′Dβλf(z) + (λ− 1)p(z)Dβ
λf(z) + p(z)Dβ+1λ f(z)
= (λ− 1)Dβ+1λ f(z) + Dβ+2
λ f(z).
Therefore, we have that
λzp(z)′Dβλf(z)Dβ+1
λ f(z) + (λ− 1)p(z)Dβ
λf(z)
Dβ+1λ f(z)
+ p(z)
= (λ− 1) +Dβ+2
λ f(z)
Dβ+1λ f(z)
,
that is, that
Dβ+2λ f(z)
Dβ+1λ f(z)
=1
p(z)
(p(z)2 + λzp′(z)
).
Therefore, we obtain
Dβ+1λ F (z)
DβλF (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),
where a ∈ C, Rea ≥ 0, β ≥ 0 and λ ≥ 1 .
119
If we denoteDβ+1
λ F (z)
DβλF (z)
= h(z), with h(0) = 1, we
have from F (z) ∈ MLβ,α(q) (see the proof of the above
Theorem):
Jβ,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) ,
where a ∈ C, Re a ≥ 0 and λ ≥ 1 .
By using the Theorem 1.6.1 and the hypothesis we
have
p(z) ≺ q(z)
orDβ+1
λ f(z)
Dβλf(z)
≺ q(z) ,
where β ≥ 0 and λ ≥ 1 .
This means f(z) = LaF (z) ∈ SL∗β(q) .
120
Remark 3.6.6 It is easy to observe that if we choose
different values for the parameters β and λ, and different
functions q(z), in the present results, we obtain every
results from the previously sections of this chapter.
121
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