Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 369078, 15 pages
doi:10.1155/2010/369078
Research Article
Differential Subordination Defined
by New Generalised Derivative Operator
for Analytic Functions
Ma’moun Harayzeh Al-Abbadi and Maslina Darus
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,
43600 Bangi Selangor (Darul Ehsan), Malaysia
Correspondence should be addressed to Maslina Darus, maslina@ukm.my
Received 9 June 2009; Revised 20 December 2009; Accepted 9 March 2010
Academic Editor: Dorothy Wallace
Copyright q 2010 M. Harayzeh Al-Abbadi and M. Darus. 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.
A new generalised derivative operator μn,m
is introduced. This operator generalised many
λ1 ,λ2
well-known operators studied earlier by many authors. Using the technique of differential
subordination, we will study some of the properties of differential subordination. In addition we
investigate several interesting properties of the new generalised derivative operator.
1. Introduction and Preliminaries
Let A denote the class of functions of the form
fz z ∞
ak zk ,
where ak is complex number,
1.1
k2
which are analytic in the open unit disc U {z ∈ C : |z| < 1} on the complex plane
C. Let S, S∗ α, Cα 0 ≤ α < 1 denote the subclasses of A consisting of functions that
are univalent, starlike of order α, and convex of order α in U, respectively. In particular, the
classes S∗ 0 S∗ and C0 C are the familiar classes of starlike and convex functions in U,
respectively. A function f ∈ Cα if Re1 zf /f > α. Furthermore a function f analytic in
U is said to be convex if it is univalent and fU is convex.
Let HU be the class of holomorphic function in unit disc U {z ∈ C : |z| < 1}.
Let
1.2
An f ∈ HU : fz z an1 zn1 · · · , z ∈ U ,
with A1 A.
2
International Journal of Mathematics and Mathematical Sciences
For a ∈ C and n ∈ N {1, 2, 3, . . .} we let
Ha, n f ∈ HU : fz z an zn an1 zn1 · · · , z ∈ U .
1.3
∞
k
k
Let fz z ∞
k2 ak z and gz z k2 bk z be analytic in the open unit disc U {z ∈
C : |z| < 1}. Then the Hadamard product or convolution f ∗ g of the two functions f, g is
defined by
∞
fz ∗ gz f ∗ g z z ak bk zk .
1.4
k2
Next, we state basic ideas on subordination. If f and g are analytic in U, then the function f
is said to be subordinate to g, and can be written as
f ≺ g,
fz ≺ gz,
z ∈ U,
1.5
if and only if there exists the Schwarz function w, analytic in U, with w0 0 and |wz| <
1 such that fz gwz, z ∈ U .
Furthermore if g is univalent in U, then f ≺ g if and only if f0 g0 and fU ⊂
gU see 1, page 36.
Let ψ : C3 × U → C and let h be univalent in U. If p is analytic in U and satisfies the
second-order differential subordination
ψ pz, zp z, z2 p z; z ≺ hz,
z ∈ U,
1.6
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 1.6. A dominant
q
that satisfies q
≺ q for all dominants q of 1.6 is said to be the best dominant of 1.6. Note
that the best dominant is unique up to a rotation of U.
Now, xk denotes the Pochhammer symbol or the shifted factorial defined by
xk ⎧
⎨1
for k 0, x ∈ C \ {0},
⎩xx 1x 2 · · · x k − 1
for k ∈ N {1, 2, 3, . . .}, x ∈ C.
1.7
To prove our results, we need the following equation throughout the paper:
n,m
n,m
μn,m1
μ
λ
fz
−
λ
fz
∗
φ
z
μ
fz
∗
φ
,
1
z
z
1
λ
1
λ
2
2
λ1 ,λ2
λ1 ,λ2
λ1 ,λ2
z ∈ U,
1.8
where n, m ∈ N0 {0, 1, 2, . . .}, λ2 ≥ λ1 ≥ 0, and φλ2 z is analytic function given by
φλ2 z z ∞
zk
.
1 λ2 k − 1
k2
1.9
International Journal of Mathematics and Mathematical Sciences
3
is the generalized derivative operator which we shall introduce later in the paper.
Here μn,m
λ1 , λ2
Moreover, we need the following lemmas in proving our main results.
Lemma 1.1 see 2, page 71. Let h be analytic, univalent, and convex in U, with h0 a, γ /
0
and, Re γ ≥ 0. If p ∈ Ha, n and
pz zp z
≺ hz,
γ
z ∈ U,
1.10
then
pz ≺ qz ≺ hz,
where qz γ/nzγ/n z
0
z ∈ U,
1.11
ht tγ/n−1 dt, z ∈ U.
The function q is convex and is the best a, n-dominant.
Lemma 1.2 see 3. Let g be a convex function in U and let
hz gz nαzg z,
1.12
where α > 0 and n is a positive integer.
If
pz g0 pn zn pn1 zn1 · · · ,
z ∈ U,
1.13
is holomorphic in U and
pz αzp z ≺ hz,
z ∈ U,
1.14
then
pz ≺ gz,
1.15
zf z
1
>− ,
Re 1 f z
2
1.16
and this result is sharp.
Lemma 1.3 see 4. Let f ∈ A, if
then
2
z
z
ftdt,
0
belongs to the class of convex functions.
z ∈ U, z / 0,
1.17
4
International Journal of Mathematics and Mathematical Sciences
2. Main Results
In the present paper, we will use the method of differential subordination to derive certain
fz. Note that differential subordination
properties of generalised derivative operator μn,m
λ1 ,λ2
has been studied by various authors, and here we follow similar works done by Oros 5 and
G. Oros and G. I. Oros 6.
In order to derive our new generalised derivative operator, we define the analytic
function
Fλm1 ,λ2 z z ∞
1 λ1 k − 1m
k2 1
λ2 k − 1
m−1
zk ,
2.1
where m ∈ N0 {0, 1, 2, . . .} and λ2 ≥ λ1 ≥ 0. Now, we introduce the new generalised
as follows.
derivative operator μn,m
λ1 ,λ2
is defined by μn,m
:A → A
Definition 2.1. For f ∈ A the operator μn,m
λ1 ,λ2
λ1 ,λ2
μn,m
fz Fλm1 ,λ2 z ∗ Rn fz,
λ1 ,λ2
z ∈ U,
2.2
where n, m ∈ N0 N ∪ {0}, λ2 ≥ λ1 ≥ 0, and Rn fz denotes the Ruscheweyh derivative
operator 7, given by
Rn fz z ∞
cn, kak zk ,
n ∈ N0 , z ∈ U,
2.3
k2
where cn, k n 1k−1 /1k−1 .
If f is given by 1.1, then we easily find from equality 2.2 that
fz z μn,m
λ1 ,λ2
∞
1 λ1 k − 1m
k2 1
λ2 k − 1m−1
cn, kak zk ,
z ∈ U,
2.4
nk−1
where n, m ∈ N0 {0, 1, 2 . . .}, λ2 ≥ λ1 ≥ 0, and cn, k n n 1k−1 /1k−1 .
Remark 2.2. Special cases of this operator include the Ruscheweyh derivative operator in two
n
≡ Rn and μn,0
≡ Rn 7, the Salagean derivative operator μ0,m
cases μn,1
1,0 ≡ S 8, the
0,λ2
λ1 ,0
generalised Ruscheweyh derivative operator in two cases μn,1
≡ Rnλ and μn,0
≡ Rnλ 9,
λ1 ,λ2
λ1 ,λ2
the generalised Salagean derivative operator μ0,m
≡ Snβ introduced by Al-Oboudi 10, and
λ1 ,0
n
≡ Dλ,β
that can be found in
the generalised Al-Shaqsi and Darus derivative operator μn,m
λ1 ,0
11.
Now, we remind the well-known Carlson-Shaffer operator La, c 12 associated with
the incomplete beta function φa, c; z, defined by
La, c : A → A,
La, cfz : φa, c; z ∗ fz,
z ∈ U,
2.5
International Journal of Mathematics and Mathematical Sciences
5
where
φa, c; z z ∞
ak−1
ck−1
k2
zk ,
2.6
a is any real number, and c /
∈ z−0 ; z−0 {0, −1, −2, . . .}.
It is easily seen that
0,1
1,2
fz μ0,m
μ0,0
0,0 fz μ0,λ2 fz μ0,1 fz La, afz fz,
λ1 ,0
2.7
1,1
0,0
μ1,0
fz μ1,m
0,0 fz μ0,λ2 fz μλ1 ,1 fz L2, 1fz zf z,
λ1 ,0
and also
fz μa−1,1
fz μa−1,m
fz La, 1fz,
μa−1,0
0,0
λ1 ,0
0,λ2
2.8
where a 1, 2, 3, . . . .
Next, we give the following.
α denote the class of
Definition 2.3. For n, m ∈ N0 , λ2 ≥ λ1 ≥ 0, and 0 ≤ α < 1, let Rn,m
λ1 ,λ2
functions f ∈ A which satisfy the condition
Re μn,m
fz
> α,
λ1 ,λ2
z ∈ U.
2.9
Also let Kλn,m
δ denote the class of functions f ∈ A which satisfy the condition
1 ,λ2
Re μn,m
fz ∗ φλ2 z > δ,
λ1 ,λ2
z ∈ U.
2.10
Remark 2.4. It is clear that R0,1
α ≡ Rλ1 , α, and the class of functions f ∈ A satisfying
λ1 ,0
Re λ1 zf z f z > α,
z ∈ U
2.11
is studied by Ponnusamy 13 and others.
Now we begin with the first result as follows.
Theorem 2.5. Let
hz 1 2α − 1z
,
1z
z ∈ U,
2.12
be convex in U, with h(0)=1 and 0 ≤ α < 1. If n, m ∈ N0 , λ2 ≥ λ1 ≥ 0, and the differential
subordination
n m1
μλ1, ,λ2 fz
≺ hz,
z ∈ U,
2.13
6
International Journal of Mathematics and Mathematical Sciences
holds, then
μn,m
fz ∗ φλ2 z
λ1 ,λ2
≺ qz 2α − 1 21 − α
1
σ
,
λ1
λ1 z1/λ1
2.14
where σ is given by
σx z
0
tx−1
dt,
1t
z ∈ U.
2.15
The function q is convex and is the best dominant.
Proof. By differentiating 1.8, with respect to z, we obtain
μn,m1
fz
λ1 ,λ2
μn,m
fz ∗ φλ2 z λ1 z μn,m
fz ∗ φλ2 z .
λ1 ,λ2
λ1 ,λ2
2.16
Using 2.16 in 2.13, differential subordination 2.13 becomes
1 2α − 1z
n,m
μn,m
.
fz
∗
φ
z
λ
z
μ
fz
∗
φ
z
≺ hz λ
1
λ
2
2
λ1 ,λ2
λ1 ,λ2
1z
2.17
Let
∞
1 λ1 k − 1m
n,m
k
pz μλ1 ,λ2 fz ∗ φλ2 z z cn, kak z
1 λ2 k − 1m
k2
1 p1 z p2 z2 · · · ,
2.18
p ∈ H1, 1, z ∈ U .
Using 2.18 in 2.17, the differential subordination becomes
pz λ1 zp z ≺ hz 1 2α − 1z
.
1z
2.19
By using Lemma 1.1, we have
pz ≺ qz 1
λ1 z1/λ1
z
ht t1/λ1 −1 dt
0
z
1 2α − 1t
t1/λ1 −1 dt
1t
0
21 − α
1
2α − 1 σ
,
1/λ
1
λ
λ1 z
1
1
λ1 z1/λ1
2.20
International Journal of Mathematics and Mathematical Sciences
7
where σ is given by 2.15, so we get
1
21 − α
n,m
μλ1 ,λ2 fz ∗ φλ2 z ≺ qz 2α − 1 σ
.
1/λ
1
λ1
λ1 z
2.21
The function q is convex and is the best dominant. The proof is complete.
Theorem 2.6. If n, m ∈ N0 , λ2 ≥ λ1 ≥ 0, and 0 ≤ α < 1, then one has
Rn,m1
⊂ Kλn,m
δ,
λ1 ,λ2 α
1 ,λ2
2.22
21 − α
1
δ 2α − 1 σ
,
λ1
λ1
2.23
where
and σ is given by 2.15.
α, then from 2.9 we have
Proof. Let f ∈ Rn,m1
λ1 ,λ2
Re μn,m1
fz
> α,
λ1 ,λ2
z ∈ U,
2.24
1 2α − 1z
.
1z
2.25
which is equivalent to
μn,m1
fz
λ1 ,λ2
≺ hz Using Theorem 2.5, we have
21 − α
1
μn,m
fz
∗
φ
z
≺
qz
2α
−
1
σ
.
λ
2
λ1 ,λ2
1/λ
λ1
λ1 z 1
2.26
Since q is convex and qU is symmetric with respect to the real axis, we deduce that
Re μn,m
fz
∗
φ
> Re q1 δ δα, λ1 z
λ
2
λ1 ,λ2
2α − 1 21 − α
1
σ
,
λ1
λ1
2.27
α ⊂ Kλn,m
δ. This completes the proof of Theorem 2.6.
from which we deduce Rn,m1
λ1 ,λ2
1 ,λ2
Remark 2.7. Special case of Theorem 2.6 with λ2 0 was given earlier in 11.
Theorem 2.8. Let q be a convex function in U, with q0 1, and let
hz qz λ1 zq z,
z ∈ U.
2.28
8
International Journal of Mathematics and Mathematical Sciences
If n, m ∈ N0 , λ2 ≥ λ1 ≥ 0,and f ∈ A and satisfies the differential subordination
μn,m1
fz
λ1 ,λ2
≺ hz,
z ∈ U,
2.29
then
μn,m
fz ∗ φλ2 z ≺ qz,
λ1 ,λ2
z ∈ U,
2.30
and this result is sharp.
Proof. Using 2.18 in 2.16, differential subordination 2.29 becomes
pz λ1 zp z ≺ hz qz λ1 zq z,
z ∈ U.
2.31
Using Lemma 1.2, we obtain
pz ≺ qz,
z ∈ U.
2.32
Hence
μn,m
fz ∗ φλ2 z ≺ qz,
λ1 ,λ2
z ∈ U.
2.33
And the result is sharp. This completes the proof of the theorem.
We give a simple application for Theorem 2.8.
Example 2.9. For n 0, m 1, λ2 ≥ λ1 ≥ 0, qz 1 z/1 − z, f ∈ A, and z ∈ U and
applying Theorem 2.8, we have
hz 1z
1 z 1 2λ1 z − z2
λ1 z
.
1−z
1−z
1 − z2
2.34
By using 1.8 we find that
μ0,1
fz 1 − λ1 fz λ1 zf z.
λ1 ,λ2
2.35
Now,
μ0,1
fz
λ1 ,λ2
∞
ak zk
ak kzk
λ1 z .
∗ φλ2 z 1 − λ1 z 1 λ2 k − 1
1 λ2 k − 1
k2
k2
∞
2.36
International Journal of Mathematics and Mathematical Sciences
9
A straightforward calculation gives the following:
∞
1 λ1 k − 1kak k−1
z
fz ∗ φλ2 z 1 μ0,1
λ1 ,λ2
1 λ2 k − 1
k2
∞
λ1 k − 1/1 λ2 k − 1zk
z
∞
fz ∗ φλ2 z ∗ z k2 k1 λ1 k − 1zk
.
z
z
k2 kak 1
2.37
Similarly, using 1.8, we find that
μ0,2
fz 1 − λ1 μ0,1
fz ∗ φλ2 z λ1 z μ0,1
fz ∗ φλ2 z ,
λ1 ,λ2
λ1 ,λ2
λ1 ,λ2
2.38
then
fz
μ0,2
λ1 ,λ2
μ0,1
fz ∗ φλ2 z λ1 z μ0,1
fz ∗ φλ2 z .
λ1 ,λ2
λ1 ,λ2
2.39
By using 2.37 we obtain
∞
kk − 1ak 1 λ1 k − 1 k−2
μ0,1
z .
fz ∗ φλ2 z λ1 ,λ2
1 λ2 k − 1
k2
2.40
We get
∞
∞
kak 1 λ1 k − 1 k−1
kk − 1ak 1 λ1 k − 1 k−1
μ0,2
fz
1
λ
z
z
1
λ1 ,λ2
1 λ2 k − 1
1 λ2 k − 1
k2
k2
1
∞
kak 1 λ1 k − 12
zk−1
1
λ
−
1
k
2
k2
2 k
fz ∗ φλ2 z ∗ z ∞
k2 k1 λ1 k − 1 z
z
2.41
.
From Theorem 2.8 we deduce that
2 k
fz ∗ φλ2 z ∗ z ∞
k2 k1 λ1 k − 1 z
z
≺
1 2λ1 z − z2
1 − z2
,
2.42
implies that
k
fz ∗ φλ2 z ∗ z ∞
1z
k2 k1 λ1 k − 1z
≺
,
z
1−z
z ∈ U.
2.43
10
International Journal of Mathematics and Mathematical Sciences
Theorem 2.10. Let q be a convex function in U, with q0 1 and let
hz qz zq z,
z ∈ U.
2.44
If n, m ∈ N0 , λ2 ≥ λ1 ≥ 0, and f ∈ A and satisfies the differential subordination
μn,m
fz
λ1 ,λ2
≺ hz,
2.45
then
μn,m
fz
λ1 ,λ2
z
≺ qz,
z ∈ U.
2.46
And the result is sharp.
Proof. Let
pz μn,m
fz
λ1 ,λ2
z
z
∞ k2
1 λ1 k − 1m /1 λ2 k − 1m−1 cn, kak zk
1 p1 z p2 z2 · · · ,
2.47
z
p ∈ H1, 1, z ∈ U .
Differentiating 2.47, with respect to z, we obtain
μn,m
fz
λ1 ,λ2
pz zp z,
z ∈ U.
2.48
Using 2.48, 2.45 becomes
pz zp z ≺ hz qz zq z.
2.49
Using Lemma 1.2, we deduce that
pz ≺ qz,
z ∈ U,
2.50
and using 2.47, we have
μn,m
fz
λ1 ,λ2
z
≺ qz,
z ∈ U.
This proves Theorem 2.10.
We give a simple application for Theorem 2.10.
2.51
International Journal of Mathematics and Mathematical Sciences
11
Example 2.11. For n 0, m 1, λ2 ≥ λ1 ≥ 0, qz 1/1 − z, f ∈ A, and z ∈ U and applying
Theorem 2.10, we have
1
1
1
z
.
1−z
1−z
1 − z2
2.52
fz 1 − λ1 fz λ1 zf z,
μ0,1
λ1 ,λ2
2.53
hz From Example 2.9, we have
so
fz
μ0,1
λ1 ,λ2
f z λ1 zf z.
2.54
Now, from Theorem 2.10 we deduce that
f z λ1 zf z ≺
1
1 − z2
2.55
implies that
1 − λ1 fz λ1 zf z
1
≺
.
z
1−z
2.56
Theorem 2.12. Let
1 2α − 1z
,
1z
hz z ∈ U,
2.57
be convex in U, with h0 1 and 0 ≤ α < 1. If n, m ∈ N0 , λ2 ≥ λ1 ≥ 0, f ∈ A, and the differential
subordination holds as
μn,m
fz
λ1 ,λ2
≺ hz,
2.58
then
μn,m
fz
λ1 ,λ2
z
≺ qz 2α − 1 The function q is convex and is the best dominant.
21 − α ln1 z
.
z
2.59
12
International Journal of Mathematics and Mathematical Sciences
Proof. Let
pz fz
μn,m
λ1 ,λ2
z
z
∞ k2
1 λ1 k − 1m /1 λ2 k − 1m−1 cn, k ak zk
2.60
z
p ∈ H1, 1, z ∈ U .
1 p1 z p2 z2 · · · ,
Differentiating 2.60, with respect to z, we obtain
fz
μn,m
λ1 ,λ2
pz zp z,
z ∈ U.
2.61
Using 2.61, the differential subordination 2.58 becomes
pz zp z ≺ hz 1 2α − 1z
,
1z
z ∈ U.
2.62
From Lemma 1.1, we deduce that
pz ≺ qz 1
z
1
z
1
z
z
htdt
0
z
0
z
0
1 2α − 1t
dt
1t
1
dt 2α − 1
1t
2α − 1 z
0
2.63
t
dt
1t
21 − α ln1 z
.
z
Using 2.60, we have
μn,m
fz
λ1 ,λ2
z
≺ qz 2α − 1 21 − α ln1 z
.
z
2.64
The proof is complete.
From Theorem 2.12, we deduce the following corollary.
Corollary 2.13. If f ∈ Rn,m
α, then
λ1 ,λ2
Re
μn,m
fz
λ1 ,λ2
z
> 2α − 1 21 − α ln 2,
z ∈ U.
2.65
International Journal of Mathematics and Mathematical Sciences
13
α, from Definition 2.3 we have
Proof. Since f ∈ Rn,m
λ1 ,λ2
Re μn,m
fz > α,
λ1 ,λ2
z ∈ U,
2.66
1 2α − 1z
.
1z
2.67
which is equivalent to
μn,m
fz
λ1 ,λ2
≺ hz Using Theorem 2.12, we have
μn,m
fz
λ1 ,λ2
≺ qz 2α − 1 21 − α
z
ln1 z
.
z
2.68
Since q is convex and qU is symmetric with respect to the real axis, we deduce that
Re
μn,m
fz
λ1 ,λ2
> Re q1 2α − 1 21 − α ln 2,
z
z ∈ U.
2.69
Theorem 2.14. Let h ∈ HU, with h0 1, h 0 /
0 which satisfy the inequality
zh z
Re 1 h z
1
>− ,
2
z ∈ U.
2.70
If n, m ∈ N0 , λ2 ≥ λ1 ≥ 0,and f ∈ A and satisfies the differential subordination
μn,m
fz
λ1 ,λ2
≺ hz,
z ∈ U,
2.71
then
μn,m
fz
λ1 ,λ2
z
≺ qz 1
z
z
htdt.
2.72
0
Proof. Let
pz fz
μn,m
λ1 ,λ2
z
z
∞ k2
1 λ1 k − 1m /1 λ2 k − 1m−1 cn, kak zk
1 p1 z p2 z2 · · · ,
z
p ∈ H1, 1, z ∈ U .
2.73
14
International Journal of Mathematics and Mathematical Sciences
Differentiating 2.73, with respect to z, we have
μn,m
fz
λ1 ,λ2
pz zp z,
z ∈ U.
2.74
Using 2.74, the differential subordination 2.71 becomes
pz zp z ≺ hz,
z ∈ U.
2.75
From Lemma 1.1, we deduce that
pz ≺ qz 1
z
z
htdt,
2.76
0
and using 2.73, we obtain
μn,m
fz
λ1 ,λ2
z
≺ qz 1
z
z
htdt.
2.77
0
From Lemma 1.3, we have that the function q is convex, and from Lemma 1.1, q is the best
dominant for subordination 2.71. This completes the proof of Theorem 2.14.
3. Conclusion
We remark that several subclasses of analytic univalent functions can be derived and studied
.
using the operator μn,m
λ1 ,λ2
Acknowledgment
This work is fully supported by UKM-ST-06-FRGS0107-2009, MOHE, Malaysia.
References
1 K. S. Padmanabhan and R. Manjini, “Certain applications of differential subordination,” Publications
de Institut Mathématique, vol. 39, no. 53, pp. 107–118, 1986.
2 S. S. Miller and P. T. Mocanu, Differential Subordinations. Theory and Applications, vol. 225 of Monographs
and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000.
3 S. S. Miller and P. T. Mocanu, “On some classes of first-order differential subordinations,” The
Michigan Mathematical Journal, vol. 32, no. 2, pp. 185–195, 1985.
4 P. T. Mocanu, T. Bulboaca, and G. S. Salagean, Teoria Geometrica a Functiilor Univalente, Casa Cartii de
Stiinta, Cluj-Napoca, Romania, 1999.
5 G. I. Oros, “A class of holomorphic functions defined using a differential operator,” General
Mathematics, vol. 13, no. 4, pp. 13–18, 2005.
6 G. Oros and G. I. Oros, “Differential superordination defined by Salagean operator,” General
Mathematics, vol. 12, no. 4, pp. 3–10, 2004.
7 S. Ruscheweyh, “New criteria for univalent functions,” Proceedings of the American Mathematical
Society, vol. 49, pp. 109–115, 1975.
International Journal of Mathematics and Mathematical Sciences
15
8 G. S. Salagean, “Subclasses of univalent functions,” in Complex Analysis—5th Romanian-Finnish
Seminar, Part 1 (Bucharest, 1981), vol. 1013 of Lecture Notes in Mathematics, pp. 362–372, Springer, Berlin,
Germany, 1983.
9 K. Al-Shaqsi and M. Darus, “On univalent functions with respect to k-symmetric points defined by
a generalized Ruscheweyh derivatives operator,” Journal of Analysis and Applications, vol. 7, no. 1, pp.
53–61, 2009.
10 F. M. Al-Oboudi, “On univalent functions defined by a generalized Salagean operator,” International
Journal of Mathematics and Mathematical Sciences, no. 25–28, pp. 1429–1436, 2004.
11 M. Darus and K. Al-Shaqsi, “Differential sandwich theorems with generalised derivative operator,”
International Journal of Computational and Mathematical Sciences, vol. 2, no. 2, pp. 75–78, 2008.
12 B. C. Carlson and D. B. Shaffer, “Starlike and prestarlike hypergeometric functions,” SIAM Journal on
Mathematical Analysis, vol. 15, no. 4, pp. 737–745, 1984.
13 S. Ponnusamy, “Differential subordination and starlike functions,” Complex Variables. Theory and
Application, vol. 19, no. 3, pp. 185–194, 1992.