Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 904272, 13 pages
doi:10.1155/2012/904272
Research Article
Sandwich-Type Theorems for a Class of
Multiplier Transformations Associated with
the Noor Integral Operators
Nak Eun Cho1 and Khalida Inayat Noor2
1
2
Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of korea
Department of Mathematics, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan
Correspondence should be addressed to Nak Eun Cho, necho@pknu.ac.kr
Received 23 September 2011; Accepted 4 November 2011
Academic Editor: Muhammad Aslam Noor
Copyright q 2012 N. E. Cho and K. I. Noor. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We obtain some subordination- and superordination-preserving properties for a class of multiplier
transformations associated with Noor integral operators defined on the space of normalized
analytic functions in the open unit disk. The sandwich-type theorems for these transformations
are also considered.
1. Introduction
Let H HU denote the class of analytic functions in the open unit disk U {z ∈ C : |z| < 1}.
For a ∈ C and nonnegative integer n, let
1.1
Ha, n f ∈ H : fz a an zn an1 zn1 · · · .
We also denote A by the subclass of Ha, 1 with the usual normalization f0 f 0 − 1 0.
Let f and F be members of H. The function f is said to be subordinate to F, or F is
said to be superordinate to f, if there exists a function w analytic in U, with w0 0 and
|wz| < 1, and such that fz Fwz. In such a case, we write f ≺ F or fz ≺ Fz. If the
function F is univalent in U, then we have f ≺ F if and only if f0 F0 and fU ⊂ FU
cf. 1.
Definition 1.1 see 1. Let φ : C2 → C and let h be univalent in U. If p is analytic in U and
satisfies the differential subordination:
φ pz, zp z ≺ hz
z ∈ U,
1.2
2
Abstract and Applied Analysis
then p is called a solution of the differential subordination. The univalent function q is called
a dominant of the solutions of the differential subordination, or more simply a dominant if
p ≺ q for all p satisfying 1.2. A dominant q that satisfies q ≺ q for all dominants q of 1.2 is
said to be the best dominant.
Definition 1.2 see 2. Let ϕ : C2 → C and let h be analytic in U. If p and ϕpz, zp z are
univalent in U and satisfy the differential superordination:
hz ≺ ϕ pz, zp z
z ∈ U,
1.3
then p is called a solution of the differential superordination. An analytic function q is called a
subordinant of the solutions of the differential superordination, or more simply a subordinant
if q ≺ p for all p satisfying 1.3. A univalent subordinant q that satisfies q ≺ q for all subordinants q of 1.3 is said to be the best subordinant.
Definition 1.3 see 2. We denote by Q the class of functions f that are analytic and injective
on U \ Ef, where
E f ζ ∈ ∂U : lim ∞ ,
z→ζ
1.4
and are such that f ζ / 0 for ζ ∈ ∂U \ Ef.
Following Komatu 3, we introduce the integral operator Lλ c : A → A defined by
cλ
L cfz :
Γλ
λ
1
0
1 λ−1
tc−2 log
ftzdt Re{c} > 0; λ ≥ 0; f ∈ A ,
t
1.5
where the symbol Γ stands the Gamma function. We also note that the operator Lλ cfz
defined by 1.5 can be expressed by the series expansion as follows:
L cfz z λ
∞ k2
c
ck−1
λ
ak zk .
1.6
Obviously, we have, for λ, ν ≥ 0,
Lλ c Lν cfz Lλν cfz.
1.7
In particular, the operator Lλ 2 is closely related to the multiplier transformation studied
earlier by Flett 4. Various interesting properties of the operator Lλ 2 have been studied by
Jung et al. 5 and Liu 6. We also note from 1.6 that we can define the operator Lλ c for
any real number λ.
Let
fcλ z z ∞ k2
c
ck−1
λ
zk
Re{c} > 0; λ ≥ 0
1.8
Abstract and Applied Analysis
λ,μ
and let fc
3
be defined such that
λ,μ
fcλ z ∗ fc z μ > 0; z ∈ U ,
z
1 − zμ
1.9
where the symbol ∗ stands for the Hadamard product or convolution. Then, motivated
essentially by the Noor integral operator 7 see also 8–11, we now introduce the operator
κ
: A → A, which are defined here by
Iλ,μ
λ,μ
λ,μ
Ic fz fc ∗ f z
f ∈ A; Re{c} > 0; λ ≥ 0; μ > 0 .
1.10
In view of 1.9 and 1.10, we obtain the following relations:
λ,μ
λ1,μ
λ,μ
fz − c − 1Ic fz,
z Ic fz cIc
1.11
λ,μ
λ,μ
λ,μ1
z Ic fz μIc
fz − μ − 1 Ic fz.
1.12
Making use of the principle of subordination between analytic functions, Miller et al.
12 investigated some subordination theorems involving certain integral operators for analytic functions in U see, also 13. Moreover, Miller and Mocanu 2 considered differential
superordinations, as the dual concept of differential subordinations see also 14. In the present paper, we obtain the subordination- and superordination-preserving properties of the
λ,μ
multiplier transformations Ic defined by 1.10 with the sandwich-type theorems.
The following lemmas will be required in our present investigation.
Lemma 1.4 see 15. Suppose that the function H : C2 → C satisfies the condition:
Re{His, t} ≤ 0,
1.13
for all real s and t ≤ −n1 s2 /2, where n is a positive integer. If the function pz 1 pn zn · · · is
analytic in U and
Re H pz, zp z > 0 z ∈ U,
1.14
then Re{pz} > 0 in U.
Lemma 1.5 see 16. Let β, γ ∈ C with β /
0 and let h ∈ HU with h0 c. If Re{βhz γ} >
0z ∈ U, then the solution of the differential equation:
qz zq z
hz
βqz γ
z ∈ U; q0 c
is analytic in U and satisfies Re{βqz γ} > 0 z ∈ U.
1.15
4
Abstract and Applied Analysis
Lemma 1.6 see 1. Let p ∈ Q with p0 a and let qz a an zn · · · be analytic in U with
qz /
≡ a and n ≥ 1. If q is not subordinate to p, then there exist points z0 r0 eiθ ∈ U and ζ0 ∈
∂U \ Ef, for which qUr0 ⊂ pU,
z0 p z0 mζ0 q ζ0 pz0 qζ0 ,
m ≥ n.
1.16
Lemma 1.7 see 2. Let q ∈ Ha, 1, let ϕ : C2 → C, and set ϕqz, zq z ≡ hz. If Lz, t ϕqz, tzq z is a subordination chain and p ∈ Ha, 1 ∩ Q, then
hz ≺ ϕ pz, zp z
z ∈ U
1.17
implies that
qz ≺ pz
z ∈ U.
1.18
Furthermore, if ϕqz, zp z hz has a univalent solution q ∈ Q, then q is the best subordinant.
A function Lz, t defined on U × 0, ∞ is the subordination chain or Löwner chain
if L·, t is analytic and univalent in U for all t ∈ 0, ∞; Lz, · is continuously differentiable
on 0, ∞ for all z ∈ U and Lz, s ≺ Lz, t for z ∈ U and 0 ≤ s < t.
0 and limt → ∞ |a1 t| ∞.
Lemma 1.8 see 17. The function Lz, t a1 tz · · · with a1 t /
Suppose that L·; t ia analytic in U for all t ≥ 0, Lz; · is continuously differentiable on 0, ∞ for all
z ∈ U. If Lz; t satisfies
|Lz; t| ≤ K0 |a1 t|
|z| < r0 < 1; 0 ≤ t < ∞
1.19
> 0 z ∈ U; 0 ≤ t < ∞,
1.20
for some positive constants K0 and r0 and
R
z∂Lz, t/∂z
∂Lz, t/∂t
then Lz; t is a subordination chain.
2. Main Results
Firstly, we begin by proving the following subordination theorem involving the multiplier
λ,μ
transformation Ic defined by 1.10.
Theorem 2.1. Let f, g ∈ A. Suppose that
zφ z
Re 1 > −δ
φ z
λ1,μ
φz : 1 − αIc
λ,μ
gz αIc gz; 0 ≤ α < 1; z ∈ U ,
2.1
Abstract and Applied Analysis
5
where
δ
1 − α2 |c − 1 α|2 − 1 − α2 − c − 1 α2 41 − α Re{c − 1 α}
Re{c − 1 α} > 0.
2.2
Then the subordination:
λ1,μ
1 − αIc
λ,μ
fz αIc fz ≺ φz
z ∈ U
2.3
implies that
λ,μ
λ,μ
Ic fz ≺ Ic gz
z ∈ U.
2.4
λ,μ
Moreover, the function Ic gz is the best dominant.
Proof. Let us define the functions F and G, respectively, by
λ,μ
Fz : Ic fz,
λ,μ
Gz : Ic gz,
2.5
We first show that if the function q is defined by
qz : 1 then
zG z
G z
Re qz > 0
z ∈ U,
z ∈ U.
2.6
2.7
Taking the logarithmic differentiation on both sides of the second equation in 2.5 and using
1.11 for g ∈ A, we obtain
cφz c − 1 αGz 1 − αzG z,
2.8
which, in conjunction with 2.8, yields the relationship:
1
zφ z
zq z
zG z
1
φ z
G z
qz c − 1 α/1 − α
zq z
qz ≡ hz.
qz c − 1 α/1 − α
2.9
From 2.1, we have
c−1α
Re hz >0
1−α
z ∈ U,
2.10
6
Abstract and Applied Analysis
and by using Lemma 1.5, we conclude that the differential equation 2.9 has a solution q ∈
HU with q0 h0 1.
Let us put
Hu, v u v
u c − 1 α/1 − α
δ,
2.11
where δ is given by 2.2. From 2.1, 2.9, and 2.11, we obtain
Re H qz, zq z > 0
z ∈ U.
2.12
Now we proceed to show that Re{His, t} ≤ 0 for all real s and t ≤ −1 s2 /2. From 2.11,
we have
t
δ
Re{His, t} Re is is c − 1 α/1 − α
tc − 1 α/1 − α
|c − 1 α/1 − α is|2
≤−
δ
Eδ s
2|c − 1 α/1 − α is|2
2.13
,
where
4δs Im{c − 1 α}
Re{c − 1 α}
Eδ s :
− 2δ s2 −
1 − α
1 − α
2
c − 1 α Re{c − 1 α} .
− 2δ
1 − α 1 − α
2.14
For δ given by 2.2, we can prove easily that the expression Eδ s given by 2.14 is positive
or equal to zero. Moreover, the quadratic expression by s in 2.14 is a perfect square for
the assumed value of δ. Hence from 2.13, we see that Re{His, t} ≤ 0 for all real s and
t ≤ −1 s2 /2. Thus, by using Lemma 1.4, we conclude that Re{qz} > 0 for all z ∈ U. That
is, q is convex in U.
Next, we prove that the subordination condition 2.3 implies that
Fz ≺ Gz
z ∈ U
2.15
for the functions F and G defined by 2.5. Without loss of generality, we can assume that G is
0 |ζ| 1. Now we consider the function Lz, t
analytic and univalent on U and that G ζ /
given by
Lz, t :
We note that
c−1α
1 − α1 t Gz zG z
c
c
z ∈ U; 0 ≤ t < ∞.
2.16
Abstract and Applied Analysis
∂Lz, t c − 1 α 1 − α1 t
0
G
0
/
∂z z0
c
7
0 ≤ t < ∞; Re{c − 1 α} > 0.
2.17
This shows that the function
Lz, t a1 tz · · ·
2.18
satisfies the condition a1 t / 0 for all t ∈ 0, ∞. By using the well-known growth and distortion theorems for convex functions, it is easy to check that the first part of Lemma 1.8 is satisfied. Furthermore, we have
Re
z∂Lz, t/∂z
∂Lz, t/∂t
Re
c−1α
zG z
1 t 1 > 0,
1−α
G z
2.19
since G is convex and Re{c − 1 α} > 0. Therefore, by virtue of Lemma 1.8, Lz, t is a subordination chain. We observe from the definition of a subordination chain that
φz 1−α c−1α
Gz zG z Lz, 0,
c
c
Lz, 0 ≺ Lz, t
2.20
z ∈ U; 0 ≤ t < ∞.
This implies that
Lζ, t ∈
/ LU, 0 φU
ζ ∈ ∂U; 0 ≤ t < ∞.
2.21
Now suppose that F is not subordinate to G, then by Lemma 1.6, there exists points
z0 ∈ U and ζ0 ∈ ∂U such that
Fz0 Gζ0 ,
z0 Fz0 1 tζ0 G ζ0 0 ≤ t < ∞.
2.22
Hence we have
c−1α
1 − α1 t
Gζ0 ζ0 G ζ0 c
c
1−α
c−1α
Fz0 z0 F z0 c
c
Lζ0 , t λ1,μ
1 − αIc
2.23
λ,μ
fz0 αIc fz0 ∈ φU,
by virtue of the subordination condition 2.3. This contracts the above observation that
/ φU. Therefore, the subordination condition 2.3 must imply the subordination
Lζ0 , t ∈
given by 2.15. Considering Fz Gz, we see that the function Gz is the best dominant.
This evidently completes the proof of Theorem 2.1.
Remark 2.2. We note that δ given by 2.2 in Theorem 2.1 satisfies the inequality 0 < δ ≤ 1/2.
8
Abstract and Applied Analysis
We next prove a dual problem of Theorem 2.1, in the sense that the subordinations are
replaced by superordinations.
Theorem 2.3. Let f, g ∈ A. Suppose that
zφ z
> −δ
Re 1 φ z
λ1,μ
φz : 1 − αIc
λ1,μ
where δ is given by 2.2, and the function Ic
superordination:
λ1,μ
φz ≺ 1 − αIc
λ,μ
gz αIc gz; 0 ≤ α < 1; z ∈ U ,
2.24
λ,μ
fz is univalent in U and Ic fz ∈ Q. Then the
λ,μ
fz αIc fz
z ∈ U
2.25
implies that
λ,μ
λ,μ
Ic gzz ≺ Ic fz
z ∈ U.
2.26
λ,μ
Moreover, the function Ic gz is the best subordinant.
Proof. Let us define the functions F and G, respectively, by 2.5. We first note that, if the function q is defined by 2.6, by using 2.8, then we obtain
1−α c−1α
Gz zG z
c
c
: ϕ Gz, zG z .
φz 2.27
After a simple calculation, 2.27 yields the relationship:
1
zq z
zφ z
.
qz
φ z
qz c − 1 α/1 − α
2.28
Then by using the same method as in the proof of Theorem 2.1, we can prove that Re{qz} >
0 for all z ∈ U. That is, G defined by 2.6 is convex univalent in U.
Next, we prove that the subordination condition 2.27 implies that
Fz ≺ Gz
z ∈ U
2.29
for the functions F and G defined by 2.5. Now consider the function Lz, t defined by
Lz, t :
c−1α
1 − αt Gz zG z
c
c
z ∈ U; 0 ≤ t < ∞.
2.30
Since G is convex and Re{c−1α} > 0, we can prove easily that Lz, t is a subordination chain
as in the proof of Theorem 2.1. Therefore according to Lemma 1.7, we conclude that the
Abstract and Applied Analysis
9
superordination condition 2.27 must imply the superordination given by 2.29. Furthermore, since the differential equation 2.27 has the univalent solution G, it is the best
subordinant of the given differential superordination. Therefore we complete the proof of
Theorem 2.3.
If we combine this Theorems 2.1 and 2.3, then we obtain the following sandwich-type
theorem.
Theorem 2.4. Let f, gk ∈ Ak 1, 2. Suppose that
zφk z
λ1,μ
λ,μ
Re 1 > −δ
φk z : 1 − αIc
gk zαIc gk z; k 1, 2; 0 ≤ α < 1; z ∈ U ,
φk z
2.31
λ1,μ
where δ is given by 2.2, and the function Ic
subordination relation:
λ1,μ
φ1 z ≺ Ic
λ,μ
fz is univalent in U and Ic fz ∈ Q. Then the
fz ≺ φ2 z
2.32
z ∈ U
implies that
λ,μ
λ,μ
λ,μ
Ic gk z ≺ Ic fz ≺ Ic g2 z
λ,μ
2.33
z ∈ U.
λ,μ
Moreover, the functions Ic g1 z and Ic g2 z are the best subordinant and the best dominant, respectively.
λ1,μ
λ,μ
fz and Ic fz need to be
The assumption of Theorem 2.4, that the functions Ic
univalent in U, may be replaced by another conditions in the following result.
Corollary 2.5. Let f, gk ∈ Ak 1, 2. Suppose that the condition 2.31 is satisfied and
zψ z
λ1,μ
λ,μ
fz αIc fz; 0 ≤ α < 1; z ∈ U; f ∈ Q ,
> −δ
ψz : 1 − αIc
Re 1 ψ z
2.34
where δ is given by 2.2. Then the subordination relation:
φ1 z ≺ ψz ≺ φ2 z
z ∈ U
2.35
implies that
λ,μ
λ,μ
λ,μ
Ic gk z ≺ Ic fz ≺ Ic g2 z
λ,μ
λ,μ
z ∈ U.
2.36
Moreover, the functions Ic g1 z and Ic g2 z are the best subordinant and the best dominant, respectively.
10
Abstract and Applied Analysis
Proof. In order to prove Corollary 2.5, we have to show that the condition 2.34 implies the
λ,μ
univalence of ψz and Fz : Ic fz. Since 0 < δ ≤ 1/2 from Remark 2.2, the condition
2.34 means that ψ is a close-to-convex function in U see 18 and hence ψ is univalent in
U. Furthermore, by using the same techniques as in the proof of Theorem 2.4, we can prove
the convexity univalence of F and so the details may be omitted. Therefore, by applying
Theorem 2.4, we obtain Corollary 2.5.
By setting c 2 and α 0 in Theorem 2.4, so that δ 1/2, we deduce the following
consequence of Theorem 2.4.
Corollary 2.6. Let f, gk ∈ Ak 1, 2. Suppose that
Re 1 λ1,μ
and the function I2
relation:
zφk z
φk z
>−
1
λ1,μ
φk z : I2
gk z; k 1, 2; z ∈ U ,
2
2.37
λ,μ
fz is univalent functions in U and I2 fz ∈ Q. Then the subordination
λ1,μ
I2
λ1,μ
g1 z ≺ I2
λ1,μ
fz ≺ I2
g2 z
z ∈ U
2.38
implies that
λ,μ
λ,μ
λ,μ
I2 g1 z ≺ I2 fz ≺ I2 g2 z
λ,μ
z ∈ U.
2.39
λ,μ
Moreover, the functions I2 g1 z and I2 g2 z are the best subordinant and the best dominant, respectively.
If we take c 2 i and α 0 in Theorem 2.4, then we easily to lead to the following
result.
Corollary 2.7. Let f, gk ∈ Ak 1, 2. Suppose that
Re 1 zφk z
φk z
√
3− 5
>−
4
λ,μ
φk z : I2i gk z; k 1, 2; z ∈ U ,
λ1,μ
2.40
λ,μ
and the function I2i fz is univalent functions in U and I2i fz ∈ Q. Then the subordination
relation:
λ1,μ
λ1,μ
λ1,μ
I2i g1 z ≺ I2i fz ≺ I2i g2 z
z ∈ U
2.41
implies that
λ,μ
λ,μ
λ,μ
I2i g1 z ≺ I2i fz ≺ I2i g2 z
z ∈ U.
2.42
Abstract and Applied Analysis
11
λ,μ
λ,μ
Moreover, the functions I2i g1 z and I2i g2 z are the best subordinant and the best dominant, respectively.
The proof of Theorem 2.8 below is similar to that of Theorem 2.4 by using 1.12 and
so the details may be omitted.
Theorem 2.8. Let f, gk ∈ A k 1, 2. Suppose that
Re 1 zφk z
> −δ
φk z
λ,μ1
φk z : 1 − αIc
λ,μ
gk zαIc gk z; k 1, 2; 0 ≤ α < 1; z ∈ U ,
2.43
λ,μ1
where δ is given by 2.2 with c μ, and the function Ic
Then the subordination relation:
λ,μ1
φ1 z ≺ Ic
λ,μ
fz is univalent in U and Ic fz ∈ Q.
fz ≺ φ2 z
z ∈ U
2.44
implies that
λ,μ
λ,μ
λ,μ
Ic gk z ≺ Ic fz ≺ Ic g2 z
λ,μ
z ∈ U.
2.45
λ,μ
Moreover, the functions Ic g1 z and Ic g2 z are the best subordinant and the best dominant, respectively.
By using a similar method given in the proof of Theorems 2.4 and 2.8, we have the
corresponding two theorems below.
Theorem 2.9. Let f, gk ∈ A k 1, 2 with the additional condition gk z / 0. Suppose that
Re 1 zφk z
φk z
λ1,μ
> −δ φk z :
1 − αIc
λ,μ
gk z αIc gk z
; k 1, 2; 0 ≤ α < 1; z ∈ U ,
z
2.46
where δ is given by
δ
λ1,μ
and the function Ic
1 − α2 |c|2 − 1 − α2 − c2 41 − α Re{c}
Re{c} > 0,
2.47
λ,μ
fz is univalent in U and Ic fz ∈ Q. Then the subordination relation:
λ1,μ
φ1 z ≺
Ic
fz
≺ φ2 z
z
z ∈ U
2.48
12
Abstract and Applied Analysis
implies that
λ,μ
λ,μ
λ,μ
Ic g1 z Ic fz Ic g2 z
≺
≺
z
z
z
λ,μ
z ∈ U.
2.49
λ,μ
Moreover, the functions Ic g1 z/z and Ic g2 z/z are the best subordinant and the best dominant,
respectively.
Theorem 2.10. Let f, gk ∈ Ak 1, 2 with the additional condition gk z / 0. Suppose that
Re 1 zφk z
φk z
λ,μ1
> −δ φk z :
1 − αIc
λ,μ
gk z αIc gk z
; k 1, 2; 0 ≤ α < 1; z ∈ U ,
z
λ,μ1
where δ is given by 2.47 with c μ, and the function Ic
fz/z ∈ Q. Then the subordination relation:
fz
≺ φ2 z
z
Ic
λ,μ
fz/z is univalent in U and Ic
λ,μ1
φ1 z ≺
2.50
z ∈ U
2.51
implies that
λ,μ
λ,μ
λ,μ
Ic g1 z Ic fz Ic g2 z
≺
≺
z
z
z
λ,μ
z ∈ U.
2.52
λ,μ
Moreover, the functions Ic g1 z/z and Ic g2 z/z are the best subordinant and the best dominant,
respectively.
Acknowledgment
This research was supported by the Basic Science Research Program through the National Research Foundation of KoreaNRF funded by the Ministry of Education, Science and Technology no. 2011–0007037.
References
1 S. S. Miller and P. T. Mocanu, Differential Subordinations, Theory and Application, vol. 225 of Monographs
and Textbooks in Pure and Applied Mathematics, Marcel Dekker Inc., New York, NY, USA, 2000.
2 S. S. Miller and P. T. Mocanu, “Subordinants of differential superordinations,” Complex Variables.
Theory and Application, vol. 48, no. 10, pp. 815–826, 2003.
3 Y. Komatu, Distortion Theorems in Relation to Linear Integral Operators, vol. 385 of Mathematics and its
Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
Abstract and Applied Analysis
13
4 T. M. Flett, “The dual of an inequality of Hardy and Littlewood and some related inequalities,” Journal
of Mathematical Analysis and Applications, vol. 38, pp. 746–765, 1972.
5 I. B. Jung, Y. C. Kim, and H. M. Srivastava, “The Hardy space of analytic functions associated with
certain one-parameter families of integral operators,” Journal of Mathematical Analysis and Applications,
vol. 176, no. 1, pp. 138–147, 1993.
6 J.-L. Liu, “A linear operator and strongly starlike functions,” Journal of the Mathematical Society of Japan,
vol. 54, no. 4, pp. 975–981, 2002.
7 K. I. Noor, “On new classes of integral operators,” Journal of Natural Geometry, vol. 16, no. 1-2, pp.
71–80, 1999.
8 J. H. Choi, M. Saigo, and H. M. Srivastava, “Some inclusion properties of a certain family of integral
operators,” Journal of Mathematical Analysis and Applications, vol. 276, no. 1, pp. 432–445, 2002.
9 J.-L. Liu, “The Noor integral and strongly starlike functions,” Journal of Mathematical Analysis and
Applications, vol. 261, no. 2, pp. 441–447, 2001.
10 J.-L. Liu and K. I. Noor, “Some properties of Noor integral operator,” Journal of Natural Geometry, vol.
21, no. 1-2, pp. 81–90, 2002.
11 K. I. Noor and M. A. Noor, “On integral operators,” Journal of Mathematical Analysis and Applications,
vol. 238, no. 2, pp. 341–352, 1999.
12 S. S. Miller, P. T. Mocanu, and M. O. Reade, “Subordination-preserving integral operators,” Transactions of the American Mathematical Society, vol. 283, no. 2, pp. 605–615, 1984.
13 T. Bulboacă, “Integral operators that preserve the subordination,” Bulletin of the Korean Mathematical
Society, vol. 34, no. 4, pp. 627–636, 1997.
14 T. Bulboacă, “A class of superordination-preserving integral operators,” Indagationes Mathematicae.
New Series, vol. 13, no. 3, pp. 301–311, 2002.
15 S. S. Miller and P. T. Mocanu, “Differential subordinations and univalent functions,” The Michigan
Mathematical Journal, vol. 28, no. 2, pp. 157–172, 1981.
16 S. S. Miller and P. T. Mocanu, “Univalent solutions of Briot-Bouquet differential equations,” Journal of
Differential Equations, vol. 56, no. 3, pp. 297–309, 1985.
17 C. Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Göttingen, Germany, 1975.
18 W. Kaplan, “Close-to-convex schlicht functions,” The Michigan Mathematical Journal, vol. 1, pp. 169–
185, 1952.
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