1_540_aouf.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 4, No. 1, 2011, 1-13

ISSN 1307-5543 – www.ejpam.com

Differential Sandwich Theorems of Analytic Functions Defined

by Linear Operators

M. K. Aouf 1,∗, Tamer M. Seoudy 2

1 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
2 Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt

Abstract. In this paper, we obtain some applications of first order differential subordination and super-

ordination results involving a linear operator and other linear operators for certain normalized analytic

functions. Some of our results generalize previously known results.

2000 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Analytic function, Hadamard product, differential subordination, superordi-

nation, linear operator

1. Introduction

Let H (U) be the class of analytic functions in the open unit disk U = {z ∈ C : |z| < 1} and

let H[a, k] be the subclass of H (U) consisting of functions of the form:

f (z) = a+ akzk + ak+1zk+1 . . . (a ∈ C). (1)

For simplicity H[a] = H[a, 1]. Also, letA be the subclass of H (U) consisting of functions of

the form:

f (z) = z +

∞
∑

k=2

akzk. (2)

If f , g ∈ H (U), we say that f is subordinate to g or f is superordinate to g, written

f (z) ≺ g(z) if there exists a Schwarz function ω, which (by definition) is analytic in U with

ω(0) = 0 and |ω(z)| < 1 for all z ∈ U , such that f (z) = g(ω(z)), z ∈ U . Furthermore, if the

function g is univalent in U , then we have the following equivalence, [cf., e.g., 6, 16, 17]:

f (z)≺ g(z)⇔ f (0) = g(0) and f (U)⊂ g(U).

∗Corresponding author.

Email addresses: mkaouf127�yahoo.
om (M. Aouf), tms00�fayoum.edu.eg (T. Seoudy)

http://www.ejpam.com 1 c© 2010 EJPAM All rights reserved.



M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 4 (2011), 1-13 2

Let φ : C2 × U → C and h(z) be univalent in U . If p (z) is analytic in U and satisfies the first

order differential subordination:

φ
�

p (z) , zp
′
(z) ; z

�

≺ h(z) , (3)

then p (z) is a solution of the differential subordination (3). The univalent function q (z) is

called a dominant of the solutions of the differential subordination (3) if p (z) ≺ q (z) for

all p (z) satisfying (3). A univalent dominant q̃ that satisfies q̃ ≺ q for all dominants of (3)

is called the best dominant. If p (z) and φ
�

p (z) , zp
′
(z) ; z

�

are univalent in U and if p (z)

satisfies first order differential superordination:

h(z) ≺ φ
�

p (z) , zp
′
(z) ; z

�

, (4)

then p (z) is a solution of the differential superordination (4). An analytic function q (z) is

called a subordinant of the solutions of the differential superordination (4) if q (z) ≺ p (z) for

all p (z) satisfying (4). A univalent subordinant q̃ that satisfies q ≺ q̃ for all subordinants of (4)

is called the best subordinant.Using the results of Miller and Mocanu [17], Bulboaca [5] con-

sidered certain classes of first order differential superordinations as well as superordination-

preserving integral operators [6]. Ali et al. [1], have used the results of Bulboaca [5] to

obtain sufficient conditions for normalized analytic functions to satisfy:

q1(z) ≺
z f ′(z)

f (z)
≺ q2(z),

where q1 and q2 are given univalent functions in U with q1(0) = q2(0) = 1. Also, Tuneski

[25] obtained a sufficient condition for starlikeness of f in terms of the quantity
f ′′(z) f (z)

( f ′(z))2
.

Recently, Shanmugam et al. [24] obtained sufficient conditions for the normalized analytic

function f to satisfy

q1(z)≺
f (z)

z f ′(z)
≺ q2(z)

and

q1(z) ≺
z2 f ′(z)

{ f (z)}2
≺ q2(z).

They [24] also obtained results for functions defined by using Carlson-Shaffer operator [7],

Ruscheweyh derivative [20] and Sălăgean operator [22].

For functions f given by (1) and g ∈ A given by g(z) = z +
∞
∑

k=2

bkzk, the Hadamard

product (or convolution) of f and g is defined by

( f ∗ g)(z) = z +

∞
∑

k=2

ak bkzk = (g ∗ f )(z). (5)

For functions f , g ∈A , we define the linear operator

Dn
λ

:A →A (λ≥ 0, n ∈ N0 = N∪ {0},N = {1,2, . . .}) by:

D0
λ( f ∗ g)(z) = ( f ∗ g)(z) ,



M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 4 (2011), 1-13 3

D1
λ( f ∗ g)(z) = Dλ( f ∗ g)(z) = (1−λ )( f ∗ g)(z) +λz ( ( f ∗ g)(z))′ , (6)

and ( in general )

Dn
λ( f ∗ g)(z) = Dλ(D

n−1
λ
( f ∗ g)(z))

= z +

∞
∑

k=2

[1+λ(k− 1)]nak bkzk
�

λ≥ 0; n ∈ N0

�

. (7)

From (7), we can easily deduce that

λz
�

Dn
λ( f ∗ g)(z)

�′
= Dn+1

λ
( f ∗ g)(z)− (1−λ)Dn

λ( f ∗ g)(z) (λ > 0). (8)

The linear operator Dn
λ
( f ∗ g)(z) was introduced by Aouf and Seoudy [3] and we observe that

Dn
λ
( f ∗ g)(z) reduces to several interesting many other linear operators considered earlier for

different choices of n, λ and the function g (z) :

(i) For bk = 1 (or g(z) =
z

1− z
), we have Dn

λ
( f ∗ g)(z) = Dn

λ
f (z), where Dn

λ
is the gener-

alized Sălăgean operator ( or Al-Oboudi operator [2] which yield Sălăgean operator Dn

for λ= 1 introduced and studied by Sălăgean [21];

(ii) For n= 0 and

g(z) = z +

∞
∑

k=2

(a1)k−1 . . . (al)k−1

(b1)k−1 . . . (bm)k−1(1)k−1

zk (9)

�

ai ∈ C; i = 1, . . . , l; b j ∈ C\Z
−
0 = {0,−1,−2, . . .} ; j = 1, . . . , m; l ≤ m+ 1; l, m ∈ N0; z ∈ U

�

,

where

(x)k =

¨

1 (k = 0; x ∈ C∗ = C\{0})
x(x + 1) . . . (x + k− 1) (k ∈ N ; x ∈ C),

we have D0
λ
( f ∗ g)(z) = ( f ∗ g)(z) = Hl ,m

�

a1; b1

�

f (z), where the operator Hl ,m

�

a1; b1

�

is the Dziok-Srivastava operator introduced and studied by Dziok and Srivastava [10]

([see also 11, 12]). The operator Hl ,m

�

a1; b1

�

, contains in turn many interesting opera-

tors such as, Hohlov linear operator (see [13]), the Carlson-Shaffer linear operator (see

[7, 21]), the Ruscheweyh derivative operator (see [20]), the Bernardi-Libera-Livingston

operator ( see [4, 14, 15]) and Owa-Srivastava fractional derivative operator (see [19]);

(iii) For n= 0 and

g(z) = z +

∞
∑

k=2

�

1+ l +λ(k− 1)

1+ l

�s

zk (λ≥ 0; l, s ∈ N0), (10)

we see that D0
λ
( f ∗ g)(z) = ( f ∗ g)(z) = I(s,λ, l) f (z), where I(s,λ, l) is the generalized

multiplier transformations which was introduced and studied by Cătaş et al. [8]. The

operator I(s,λ, l), contains as special cases, the multiplier transformation I(s, l) (see

[9]) for λ = 1, the generalized Sălăgean operator Dn
λ

introduced and studied by Al-

Oboudi [2] which in turn contains as special case the Sălăgean operator Dn (see [21]);



M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 4 (2011), 1-13 4

(iv) For g(z) of the form (9), the operator Dn
λ
( f ∗ g)(z) = Dn

λ
(a1, b1) f (z), introduced and

studied by Selvaraj and Karthikeyan [23].

In this paper, we will derive several subordination results, superordination results and

sandwich results involving the operator Dn
λ
( f ∗ g)(z) and some of its special operators by

some choices of n,λ and the function g(z).

2. Preliminaries

In order to prove our subordinations and superordinations, we need the following defini-

tion and lemmas.

Definition 1. [17] Denote by Q, the set of all functions f that are analytic and injective on

U\E( f ), where

E( f ) =

�

ζ ∈ ∂ U : lim
z→ζ

f (z) =∞
�

,

and are such that f
′
(ζ) 6= 0 for ζ ∈ ∂ U\E

�

f
�

.

Lemma 1. [17] Let q (z) be univalent in the unit disk U and θ and ϕ be analytic in a domain

D containing q(U) with ϕ (w) 6= 0 when w ∈ q(U). Set

ψ (z) = zq
′
(z)ϕ

�

q (z)
�

and h(z) = θ
�

q (z)
�

+ψ (z) . (11)

Suppose that

(i) ψ (z) is starlike univalent in U,

(ii) ℜ

(

zh
′
(z)

ψ (z)

)

> 0 for z ∈ U.

If p (z) is analytic with p(0) = q(0), p(U)⊂ D and

θ
�

p (z)
�

+ zp
′
(z)ϕ

�

p (z)
�

≺ θ
�

q (z)
�

+ zq
′
(z)ϕ

�

q (z)
�

, (12)

then p(z) ≺ q(z) and q (z) is the best dominant.

Taking θ (w) = αw and ϕ (w) = γ in Lemma 1, Shanmugam et al. [24] obtained the

following lemma.

Lemma 2. [24] Let q (z) be univalent in U with q(0) = 1. Let α ∈ C; γ ∈ C∗, further assume

that

ℜ

(

1+
zq
′′
(z)

q
′
(z)

)

>max

�

0,−ℜ
�

α

γ

��

. (13)

If p (z) is analytic in U, and

αp (z) + γzp
′
(z) ≺ αq (z) + γzq

′
(z) ,

then p (z)≺ q (z) and q (z) is the best dominant.



M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 4 (2011), 1-13 5

Lemma 3. [5] Let q (z) be convex univalent in U and ϑ and φ be analytic in a domain D

containing q(U). Suppose that

(i) ℜ
�

ϑ
′
(q(z))

φ(q(z))

�

> 0 for z ∈ U,

(ii) Ψ(z) = zq
′
(z)φ

�

q (z)
�

is starlike univalent in U.

If p(z) ∈ H[q(0), 1]∩Q, with p(U) ⊆ D, and ϑ
�

p (z)
�

+ zp
′
(z)φ

�

p (z)
�

is univalent in U

and

ϑ
�

q (z)
�

+ zq
′
(z)φ

�

q (z)
�

≺ ϑ
�

p (z)
�

+ zp
′
(z)φ

�

p (z)
�

, (14)

then q(z)≺ p(z) and q (z) is the best subordinant.

Taking ϑ (w) = αw and φ (w) = γ in Lemma 3, Shanmugam et al. [24] obtained the

following lemma.

Lemma 4. [24] Let q (z) be convex univalent in U , q(0) = 1. Let α ∈ C; γ ∈ C∗ and

ℜ
�

α

γ

�

> 0. If p(z) ∈ H[q(0), 1]∩Q, αp (z) + γzp
′
(z) is univalent in U and

αq (z) + γzq
′
(z)≺ αp (z) + γzp

′
(z) ,

then q (z) ≺ p (z) and q (z) is the best subordinant.

3. Sandwich Results

Unless otherwise mentioned, we assume throughout this paper that λ > 0 and n ∈ N0.

Theorem 1. Let q (z) be univalent in U with q(0) = 1, and γ ∈ C∗. Further, assume that

ℜ

(

1+
zq
′′
(z)

q
′
(z)

)

>max

�

0,−ℜ
�

1

γ

��

. (15)

If f , g ∈A satisfy the following subordination condition:

�

1+
γ

λ

� zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
+
γ

λ







zDn+2
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
− 2

z
�

Dn+1
λ
( f ∗ g)(z)

�2

�

Dn
λ
( f ∗ g)(z)

�3







≺ q (z) + γzq
′
(z) , (16)

then
zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
≺ q (z)

and q (z) is the best dominant.



M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 4 (2011), 1-13 6

Proof. Define a function p (z) by

p (z) =
zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
(z ∈ U) . (17)

Then the function p (z) is analytic in U and p(0) = 1. Therefore, differentiating (17) logarith-

mically with respect to z and using the identity (8) in the resulting equation, we have

�

1+
γ

λ

� zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
+
γ

λ







zDn+2
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
− 2

z
�

Dn+1
λ
( f ∗ g)(z)

�2

�

Dn
λ
( f ∗ g)(z)

�3







= p (z)+γzp
′
(z) ,

that is,

p (z) + γzp
′
(z) ≺ q (z) + γzq

′
(z) .

Therefore, Theorem 1 now follows by applying Lemma 2.

Putting q(z) =
1+ Az

1+ Bz
(−1≤ B < A≤ 1) in Theorem 1, we obtain the following corollary.

Corollary 1. Let γ ∈ C∗ and

ℜ
�

1− Bz

1+ Bz

�

>max

�

0,−ℜ
�

1

γ

��

.

If f , g ∈A satisfy the following subordination condition:

�

1+
γ

λ

� zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
+
γ

λ







zDn+2
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
− 2

z
�

Dn+1
λ
( f ∗ g)(z)

�2

�

Dn
λ
( f ∗ g)(z)

�3







≺
1+A z

1+ Bz
+ γ
(A− B) z

(1+ Bz)2
,

then
zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
≺

1+ Az

1+ Bz

and the function
1+ Az

1+ Bz
is the best dominant.

Remark 1. Taking g(z) =
z

1− z
in Theorem 1, we obtain the subordination result of Nechita

[18, Theorem 14].

Remark 2. Taking λ = 1 and g(z) =
z

1− z
in Theorem 1, we obtain the subordination result

for Sălăgean operator which was obtained by Shanmugam et al. [24, Theorem 5.4] and also

obtained by Nechita [18, Corollary 16].



M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 4 (2011), 1-13 7

Taking n = 0,λ = 1 and g (z) of the form (9) in Theorem 1, we obtain the following

subordination result for Dziok-Srivastava operator.

Corollary 2. Let q (z) be univalent in U with q(0) = 1, and γ ∈ C∗. Further assume that (15)

holds. If f ∈A satisfies the following subordination condition:

z2
�

Hl ,m

�

a1; b1

�

f (z)
�′

�

Hl ,m

�

a1; b1

�

f (z)
�2
− γz2

 

z
�

Hl ,m

�

a1; b1

�

f (z)
�

!′′

≺ q (z) + γzq
′
(z) ,

then

z2
�

Hl ,m

�

a1; b1

�

f (z)
�′

�

Hl ,m

�

a1; b1

�

f (z)
�2
≺ q (z)

and q (z) is the best dominant.

Taking g (z) of the form (9) in Theorem 1, we obtain the following subordination result

for the operator Dn
λ
(a1; b1).

Corollary 3. Let q (z) be univalent in U with q(0) = 1, and γ ∈ C∗. Further assume that [15]

holds. If f ∈A satisfies the following subordination condition:

�

1+
γ

λ

� zDn+1
λ
(a1; b1) f (z))

�

Dn
λ
(a1; b1) f (z)

�2
+
γ

λ







zDn+2
λ
(a1; b1) f (z)

�

Dn
λ
(a1; b1) f (z)

�2
− 2

z
�

Dn+1
λ
(a1; b1) f (z)

�2

�

Dn
λ
(a1; b1) f (z)

�3







≺ q (z) + γzq
′
(z) ,

then
zDn+1
λ
(a1; b1) f (z))

�

Dn
λ
(a1; b1) f (z)

�2
≺ q (z)

and q (z) is the best dominant.

Taking n= 0,λ= 1 and

g(z) = z +

∞
∑

k=2

�

l + k

1+ l

�s

zk (l, s ∈ N0), (18)

in Theorem 1, we obtain the following subordination result for the multiplier transformations

I(s, l).

Corollary 4. Let q (z) be univalent in U with q(0) = 1, and γ ∈ C∗. Further assume that (15)

holds. If f ∈A satisfies the following subordination condition:

z2
�

I(s, l) f (z)
�′

�

I(s, l) f (z)
�2
− γz2

�

z

I(s, l) f (z)

�′′

≺ q (z) + γzq
′
(z) ,



M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 4 (2011), 1-13 8

then

z2
�

I(s, l) f (z)
�′

�

I(s, l) f (z)
�2
≺ q (z)

and q (z) is the best dominant.

Remark 3. Taking n = 0,λ = 1 and g(z) =
z

1− z
in Theorem 1, we obtain the subordination

result of Shanmugam et al. [24, Theorem 3.4] and also obtained by Nechita [18, Corollary 17].

Now, by appealing to Lemma 4 it can be easily prove the following theorem.

Theorem 2. Let q (z) be convex univalent in U with q (0) = 1. Let γ ∈ C with ℜ
�

γ
�

> 0. If

f , g ∈A ,
zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
∈ H [1,1]∩Q,

�

1+
γ

λ

� zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
+
γ

λ







zDn+2
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
− 2

z
�

Dn+1
λ
( f ∗ g)(z)

�2

�

Dn
λ
( f ∗ g)(z)

�3







is univalent in U, and the following superordination condition

q (z)+γzq
′
(z) ≺

�

1+
γ

λ

� zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
+
γ

λ







zDn+2
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
− 2

z
�

Dn+1
λ
( f ∗ g)(z)

�2

�

Dn
λ
( f ∗ g)(z)

�3







holds, then

q (z) ≺
zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2

and q (z) is the best subordinant.

Taking q(z) =
1+ Az

1+ Bz
(−1≤ B < A≤ 1) in Theorem 2, we have the following corollary.

Corollary 5. Let γ ∈ C with ℜ
�

γ
�

> 0. If f , g ∈A ,
zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
∈ H [1,1]∩Q,

�

1+
γ

λ

� zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
+
γ

λ







zDn+2
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
− 2

z
�

Dn+1
λ
( f ∗ g)(z)

�2

�

Dn
λ
( f ∗ g)(z)

�3







is univalent in U, and the following superordination condition

1+ Az

1+ Bz
+γ
(A− B) z

(1+ Bz)2
≺
�

1+
γ

λ

� zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
+
γ

λ







zDn+2
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
− 2

z
�

Dn+1
λ
( f ∗ g)(z)

�2

�

Dn
λ
( f ∗ g)(z)

�3









M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 4 (2011), 1-13 9

holds, then

1+ Az

1+ Bz
≺

zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2

and q (z) is the best subordinant.

Remark 4. Taking g(z) =
z

1− z
in Theorem 2, we obtain the superordination result of Nechita

[18, Theorem 19].

Remark 5. Taking λ = 1 and g(z) =
z

1− z
in Theorem 2, we obtain the following superordina-

tion result for Sălăgean operator which is obtained Shanmugam et al. [24, Theorem 5.5].

Taking n = 0,λ = 1 and g (z) of the form (9) in Theorem 2, we obtain the following

superordination result for Dziok-Srivastava operator.

Corollary 6. Let q (z) be convex univalent in U with q (0) = 1. Let γ ∈ C with ℜ
�

γ
�

> 0. If

f ∈A ,
z2
�

Hl ,m

�

a1; b1

�

f (z)
�′

�

Hl ,m

�

a1; b1

�

f (z)
�2
∈ H [1,1]∩Q,

z2
�

Hl ,m

�

a1; b1

�

f (z)
�′

�

Hl ,m

�

a1; b1

�

f (z)
�2
− γz2

 

z
�

Hl ,m

�

a1; b1

�

f (z)
�

!′′

is univalent in U, and the following superordination condition

q (z) + γzq
′
(z)≺

z2
�

Hl ,m

�

a1; b1

�

f (z)
�′

�

Hl ,m

�

a1; b1

�

f (z)
�2
− γz2

 

z
�

Hl ,m

�

a1; b1

�

f (z)
�

!′′

holds, then

q (z) ≺
z2
�

Hl ,m

�

a1; b1

�

f (z)
�′

�

Hl ,m

�

a1; b1

�

f (z)
�2

and q (z) is the best subordinant.

Taking g (z) of the form (9) in Theorem 2, we obtain the following superordination result

for the operator Dn
λ
(a1; b1).

Corollary 7. Let q (z) be convex univalent in U with q (0) = 1. Let γ ∈ C with ℜ
�

γ
�

> 0. If

f , g ∈A ,
zDn+1
λ
(a1; b1) f (z)

�

Dn
λ
(a1; b1) f (z)

�2
∈ H [1,1]∩Q,

�

1+
γ

λ

� zDn+1
λ
(a1; b1) f (z)

�

Dn
λ
(a1; b1) f (z)

�2
+
γ

λ







zDn+2
λ
(a1; b1) f (z)

�

Dn
λ
(a1; b1) f (z)

�2
− 2

z
�

Dn+1
λ
(a1; b1) f (z)

�2

�

Dn
λ
(a1; b1) f (z)

�3









M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 4 (2011), 1-13 10

is univalent in U, and the following superordination condition

q (z)+γzq
′
(z) ≺

�

1+
γ

λ

� zDn+1
λ
(a1; b1) f (z)

�

Dn
λ
(a1; b1) f (z)

�2
+
γ

λ







zDn+2
λ
(a1; b1) f (z)

�

Dn
λ
(a1; b1) f (z)

�2
− 2

z
�

Dn+1
λ
(a1; b1) f (z)

�2

�

Dn
λ
(a1; b1) f (z)

�3







holds, then

q (z) ≺
zDn+1
λ
(a1; b1) f (z)

�

Dn
λ
(a1; b1) f (z)

�2

and q (z) is the best subordinant.

Taking n = 0,λ = 1 and g(z) of the form (18) in Theorem 2, we obtain the following

supordination result for the multiplier transformations I(s, l).

Corollary 8. Let q (z) be convex univalent in U with q (0) = 1. Let γ ∈ C with ℜ
�

γ
�

> 0. If

f ∈A ,
z2
�

I(s, l) f (z)
�′

�

I(s, l) f (z)
�2
∈ H [1,1]∩Q,

z2
�

I(s, l) f (z)
�′

�

I(s, l) f (z)
�2
− γz2

�

z

I(s, l) f (z)

�′′

is univalent in U, and the following superordination condition

q (z) + γzq
′
(z)≺

z2
�

I(s, l) f (z)
�′

�

I(s, l) f (z)
�2
− γz2

�

z

I(s, l) f (z)

�′′

holds, then

q (z) ≺
z2
�

I(s, l) f (z)
�′

�

I(s, l) f (z)
�2

and q (z) is the best subordinant.

Remark 6. Taking n= 0,λ= 1 and g(z) =
z

1− z
in Theorem 2, we obtain the superordination

result of Shanmugam et al. [24, Theorem 3.5].

Combining Theorem 1 and Theorem 2, we get the following sandwich theorem for the

linear operator Dn
λ
( f ∗ g).

Theorem 3. Let q1 (z) be convex univalent in U with q1 (0) = 1, γ ∈ C with ℜ
�

γ
�

> 0,q2 (z) be

univalent in U with q2 (0) = 1, and satisfies (15). If f , g ∈A ,
zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
∈ H [1,1]∩Q,

�

1+
γ

λ

� zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
+
γ

λ







zDn+2
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
− 2

z
�

Dn+1
λ
( f ∗ g)(z)

�2

�

Dn
λ
( f ∗ g)(z)

�3









M. Aouf, T. Seoudy / Eur. J. Pure Appl. Math, 4 (2011), 1-13 11

is univalent in U, and

q1 (z) + γzq
′

1 (z)

≺
�

1+
γ

λ

� zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
+
γ

λ







zDn+2
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
− 2

z
�

Dn+1
λ
( f ∗ g)(z)

�2

�

Dn
λ
( f ∗ g)(z)

�3







≺ q2 (z) + γzq
′

2 (z)

holds, then

q1 (z) ≺
zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
≺ q2 (z)

and q1 (z) and q2 (z) are, respectively, the best subordinant and the best dominant.

Taking qi(z) =
1+ Aiz

1+ Biz

�

i = 1,2;−1≤ B2 ≤ B1 < A1 ≤ A2 ≤ 1
�

in Theorem 3, we have the

following corollary.

Corollary 9. Let γ ∈ C with ℜ
�

γ
�

> 0. If f , g ∈A ,
zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
∈ H [1,1]∩Q,

�

1+
γ

λ

� zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
+
γ

λ







zDn+2
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
− 2

z
�

Dn+1
λ
( f ∗ g)(z)

�2

�

Dn
λ
( f ∗ g)(z)

�3







is univalent in U, and

1+ A1z

1+ B1z
+ γ

�

A1 − B1

�

z
�

1+ B1z
�2

≺
�

1+
γ

λ

� zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
+
γ

λ







zDn+2
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
− 2

z
�

Dn+1
λ
( f ∗ g)(z)

�2

�

Dn
λ
( f ∗ g)(z)

�3







≺
1+ A2z

1+ B2z
+ γ

�

A2 − B2

�

z
�

1+ B2z
�2

holds, then

1+ A1z

1+ B1z
≺

zDn+1
λ
( f ∗ g)(z)

�

Dn
λ
( f ∗ g)(z)

�2
≺

1+ A2z

1+ B2z

and
1+ A1z

1+ B1z
and

1+A2z

1+ B2z
are, respectively, the best subordinant and the best dominant.



REFERENCES 12

Remark 7. Taking g(z) =
z

1− z
in Theorem 3, we obtain sandwich result of Nechita [18,

Theorem 19].

Remark 8. Taking λ = 1 and g(z) =
z

1− z
in Theorem 3, we obtain sandwich result of Shan-

mugam et al. [24, Theorem 5.6].

Remark 9. Combining (i) Corollary 2 and Corollary 6; (ii) Corollary 3 and Corollary 7; (iii)

Corollary 4 and Corollary 8, we obtain similar sandwich theorems for the corresponding linear

operators.

Remark 10. Taking n= 0,λ= 1 and g(z) =
z

1− z
in Theorem 3, we obtain the sandwich result

of Shanmugam et al. [24, Corollary 3.6].

References

[1] R. M. Ali, V. Ravichandran, and K. G. Subramanian. Differential sandwich theorems for

certain analytic functions. Far East J. Math. Sci., 15(1):87–94, 2004.

[2] F. M. AlOboudi. On univalent functions defined by a generalized Salagean operator.

Internat. J. Math. Math. Sci., 27:1429–1436, 2004.

[3] M. K. Aouf and T. M. Seoudy. On differential sandwich theorems of analytic functions

defined by certain linear operator. Ann. Univ. Mariae Curie-Sklodowska Sect. A, (To

appear).

[4] S. D. Bernardi. Convex and starlike univalent functions. Trans. Amer. Math. Soc.,

135:429–446, 1969.

[5] T. Bulboaca. Classes of first order differential superordinations. Demonstratio Math.,

35(2):287–297, 2002.

[6] T. Bulboaca. Differential Subordinations and Superordinations, Recent Results. House of

Scientific Book Publ., Cluj-Napoca, 2005.

[7] B. C. Carlson and D. B. Shaffer. Starlike and prestarlike hypergeometric functions. SIAM

J. Math. Anal., 15:737–745, 1984.

[8] A. Catas, G. I. Oros, and G. Oros. Differential subordinations associated with multiplier

transformations. Abstract Appl. Anal., 2008,ID 845724:1–11, 2008.

[9] N. E. Cho and T. G. Kim. Multiplier transformations and strongly close-to-convex func-

tions. Bull. Korean Math. Soc., 40(3):399–410, 2003.

[10] J. Dziok and H. M. Srivastava. Classes of analytic functions associated with thegeneral-

ized hypergeometric function. Appl. Math. Comput., 103:1–13, 1999.



REFERENCES 13

[11] J. Dziok and H. M. Srivastava. Some subclasses of analytic functions with fixed argu-

ment of coefficients associated with the generalized hypergeometric function. Adv. Stud.

Contemp. Math., 5:115–125, 2002.

[12] J. Dziok and H. M. Srivastava. Certain subclasses of analytic functions associated with

the generalized hypergeometric function. Integral Transform. Spec. Funct., 14:7–18,

2003.

[13] Yu. E. Hohlov. Operators and operations in the univalent functions. Izv. Vyŝsh. Učebn.

Zaved. Mat.( in Russian), 10:83–89, 1978.

[14] R. J. Libera. Some classes of regular univalent functions. Proc. Amer. Math. Soc., 16:755–

658, 1965.

[15] A. E. Livingston. On the radius of univalence of certain analytic functions. Proc. Amer.

Math. Soc., 17:352–357, 1966.

[16] S. S. Miller and P. T. Mocanu. Differential Subordination: Theory and Applications, Series

on Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225. Marcel Dekker

Inc., New York and Basel, 2000.

[17] S. S. Miller and P. T. Mocanu. Subordinates of differential superordinations. Complex

Variables, 48(10):815–826, 2003.

[18] V. O. Nechita. Differential subordinations and superordinations for analytic functions

defined by the generalized Sălăgean derivative. Acta Univ. Apulensis, 16:143–156, 2008.

[19] S. Owa and H. M. Srivastava. Univalent and starlike generalized hypergeometric func-

tions. Canad. J. Math., 39:1057–1077, 1987.

[20] St. Ruscheweyh. New criteria for univalent functions. Proc. Amer. Math. Sco., 49:109–

115, 1975.

[21] H. Saitoh. A linear operator ana its applications of fiest order differential subordinations.

Math. Japon., 44:31–38, 1996.

[22] G. S. Salagean. Subclasses of univalent functions. Lecture Notes in Math. (Springer-

Verlag), 1013:362–372, 1983.

[23] C. Selvaraj and K. R. Karthikeyan. Differential subordination and superordination for

certain subclasses of analytic functions. Far East J. Math. Sci., 29(2):419–430, 2008.

[24] T. N. Shanmugam, V. Ravichandran, and S. Sivasubramanian. Differantial sandwich

theorems for some subclasses of analytic functions. J. Austr. Math. Anal. Appl., 3(1, Art.

8):1–11, 2006.

[25] N. Tuneski. On certain sufficient conditions for starlikeness. Internat. J. Math. Math.

Sci., 23(8):521–527, 2000.