3_597_aouf.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 4, 2010, 641-652

ISSN 1307-5543 – www.ejpam.com

Some Sandwich Theorems for Certain Analytic Functions

Defined by Convolution

M. K. Aouf∗ and A. O. Mostafa

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

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

perordination results for some analytic functions defined by convolution.

2000 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Analytic functions, differential subordination , superordination, sandwich

theorems, convolution.

1. Introduction

Let S denote the class of functions of the form:

f (z) = z +

∞∑

k=2

akzk, (1)

which are analytic and univalent in the open unit disk U = {z : z ∈ C , |z| < 1} . If f and g

are analytic functions in U , we say that f is subordinate to g, written f ≺ g if there exists a

Schwarz function w, which (by definition) is analytic in U with w(0) = 0 and |w(z)| < 1 for

all z ∈ U , such that f (z) = g(w(z)), z ∈ U . Furthermore, if the function g is univalent in U ,

then we have the following equivalence:

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

Let H(U) denote the class of analytic functions in U and let H[a, 1] denote the subclass of the

functions f ∈ H(U) of the form:

f (z) = a+ a1z + a2z2 + . . . (a ∈ C ).

∗Corresponding author.

Email addresses: mkaouf127�yahoo.
om (M. Aouf), adelaeg254�yahoo.
om (A. Mostafa)

http://www.ejpam.com 641 c© 2010 EJPAM All rights reserved.



M. Aouf and A. Mostafa / Eur. J. Pure Appl. Math, 3 (2010), 641-652 642

Supposing that h and g are two analytic functions in U, let

ϕ(r, s, t; z) : C3 ×U→ C .

If h and ϕ(h(z), zh′(z), z2h
′′
(z); z) are univalent functions in U and if h satisfies the second-

order superordination

g(z) ≺ ϕ(h(z), zh′(z), z2h
′′
(z); z), (2)

then g is a solution of the differential superordination (2). A function g ∈ H(U) is called a

subordinant of (2), if q(z) ≺ h(z) for all the functions h satisfying (2). A univalent subor-

dinant eq that satisfies q(z) ≺ eq(z) for all of the subordinants q of (2), is said to be the best

subordinant.

Recently, Miller and Mocanu [15] obtained sufficient conditions on the functions g, q and

ϕ for which the following implication holds:

g(z) ≺ ϕ(h(z), zh′(z), z2h
′′
(z); z)⇒ q(z) ≺ h(z).

Using the results of Miller and Mocanu [15], Bulboaca [4] considered certain classes of first

order differential superordinations as well as superordination-preserving integral operators

[5]. Ali et al. [1], have used the results of Bulboaca [4] 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 normalized functions in U.

Very recently, Shanmugam et al. [23] obtained sufficient conditions for a 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) ,

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

For functions f given by (1) and g ∈ S 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). (3)

We observe that for different choices of the function g, the function ( f ∗ g)(z) reduces to

several interesting operators. For example, if

g(z) = z +

∞∑

k=2

(a)k−1

(c)k−1

zk (c 6= 0,−1,−2, ...; z ∈ U), (4)



M. Aouf and A. Mostafa / Eur. J. Pure Appl. Math, 3 (2010), 641-652 643

where

(d)k =

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

we see that, ( f ∗ g)(z) = L(a, c) f (z) and L(a, c) is the Carlson-Shaffer operator [6]. If

g(z) = z +

∞∑

k=2

(α1)k−1...(αl)k−1

(β1)k−1...(βs)k−1(1)k−1

zk, (5)

where, αi > 0 (i = 1,2, ...l);β j > 0 ( j = 1,2, ...s), l ≤ s + 1, l, s ∈ N0 = N ∪ {0}, where

N = {1,2, ...}, we see that, ( f ∗ g)(z) = Hl ,s(α1) f (z), where Hl ,s(α1) is the Dziok-Srivastava

operator introduced and studied by Dziok and Srivastava [9] ( see also [10] and [11]). The

operator Hl ,s(α1), contains in tern many interesting operators such as, Hohlov linear operator

(see [12]), the Carlson-Shaffer linear operator (see [6] and [21] ), the Ruscheweyh derivative

operator (see [20]), the Bernardi-Libera-Livingston operator ( see [13]) and Owa-Srivastava

fractional derivative operator (see [18]).

Also, if

g(z) = z +

∞∑

k=2

�
1+ l +λ(k− 1)

1+ l

�m
zk (λ¾ 0, l ¾ 0, m ∈ N0), (6)

we see that ( f ∗ g)(z) = I(m,λ, l) f (z), where I(m,λ, l) is the generalized multiplier trans-

formation which was introduced and studied by Cătaş et al. [7]. The operator I(m,λ, l),

contains as special cases, the multiplier transformation (see [8]), the generalized Salăgeăn

operator introduced and studied by Al-Oboudi [2] which in tern contains as special case the

Salăgeăn operator (see [22]).

In [16], Mostafa et al. obtained some interesting subordination results for the function�
( f ∗ g)(z)

z

�α
(α ∈ C∗).

In this paper, we get some interesting subordination results for the function�
z

( f ∗ g)(z)

�δ
(δ ∈ C∗).

2. Definitions and Preliminaries

To prove our results we shall need the following definition and lemmas.

Definition 1 ([15]). Let Q be 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 ([14]). Let q be univalent in the unit disc U, and let 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)), h(z) =

θ(q(z)) +ψ(z) and suppose that



M. Aouf and A. Mostafa / Eur. J. Pure Appl. Math, 3 (2010), 641-652 644

(i) ψ is a starlike function in U,

(ii) Re
zh′(z)

ψ(z)
> 0, z ∈ U.

If p is analytic in U 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)), (7)

then p(z) ≺ q(z), and q is the best dominant of (7).

Lemma 2 ([23]). Let µ,γ ∈ C∗, and let q be a convex function in U with

Re

�
1+

zq′′(z)

q′(z)
+
µ

γ

�
> 0 , z ∈ U.

If p is analytic in U and

µp(z) + γzp′(z) ≺ µq(z) + γzq′(z), (8)

then p(z) ≺ q(z), and q is the best dominant of (8).

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

D containing q(U). Suppose that:

(i) Re
θ ′(q(z))

ϕ(q(z))
> 0, z ∈ U,

(ii) h(z) = zq′(z)ϕ(q(z)) is starlike in U.

If p ∈ H[q(0), 1]∩Q with p(U)⊂ D, the function 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)), (9)

then q(z)≺ p(z), and q is the best subordinant of (9).

Lemma 4 ([19]). The function q(z) = (1− z)−2ab is univalent in U if and only if |2ab− 1| ≤ 1

or |2ab+ 1| ≤ 1.

3. Main Results

Unless otherwise mentioned, we assume throughout this paper that, δ,η ∈ C∗, z ∈ U and

the power is the principal one.

Theorem 1. Let q be univalent in U and satisfies

Re{1+
zq′′(z)

q′(z)
+
δ

η
}> 0. (10)

If f , g ∈ S with ( f ∗ g)(z) 6= 0, z ∈ U∗ = U\{0} satisfy the subordination:

χg(η,δ, f )≺ q(z) +
η

δ
zq′(z), (11)



M. Aouf and A. Mostafa / Eur. J. Pure Appl. Math, 3 (2010), 641-652 645

where χg(η,δ, f ) is given by

χg(η,δ, f ) = (1+η)

�
z

( f ∗ g)(z)

�δ
−η

z
�
( f ∗ g)(z)
�′

( f ∗ g)(z)

�
z

( f ∗ g)(z)

�δ
, (12)

then

(
z

( f ∗ g)(z)
)δ ≺ q(z) (13)

and q is the best dominant.

Proof. Define a function p by

p(z) = (
z

( f ∗ g)(z)
)δ. (14)

Then the function p is analytic in U and p(0) = 1. Therefore, by differentiating (14) logarith-

mically with respect to z, we have

p(z) +
η

δ
zp′(z) = (1+η)

�
z

( f ∗ g)(z)

�δ
−η

z
�
( f ∗ g)(z)
�′

( f ∗ g)(z)

�
z

( f ∗ g)(z)

�δ
. (15)

Using (11) and (15), we have

p(z) +
η

δ
zp′(z) ≺ q(z) +

η

δ
zq′(z). (16)

Hence, the assertion (13) now follows by using Lemma 2 with γ =
η

δ
and µ = 1.

Putting q(z) = (1+ Az)/(1+ Bz) (−1 ≤ B < A ≤ 1) in Theorem 1, the condition (10)

becomes

Re

�
1− Bz

1+ Bz
+
δ

η

�
> 0, z ∈ U . (17)

It is easy to check that the function φ(z) =
1−ζ
1+ζ

, |ζ| < |B| ≤ 1, is convex in U , and since

φ(ζ) = φ(ζ) for all |ζ| < |B| , it follows that the image φ(U) is a convex domain symmetric

with respect to the real axis, hence

inf

�
Re

1− Bz

1+ Bz

�
=

1− |B|

1+ |B|
¾ 0.

Then, the inequality (17) is equivalent to

Re
η

δ
¾
|B| − 1

1+ |B|
, (18)

hence, we have the following corollary.



M. Aouf and A. Mostafa / Eur. J. Pure Appl. Math, 3 (2010), 641-652 646

Corollary 1. Let −1≤ B < A≤ 1 and (18) holds. If f (z) ∈ S with ( f ∗ g)(z) 6= 0, z ∈ U∗ and

χg(η,δ, f )≺
1+ Az

1+ Bz
+
η

δ

(A− B)z

(1+ Bz)2
,

where χg(η,δ, f ) is given by (12), then

�
z

( f ∗ g)(z)

�δ
≺

1+ Az

1+ Bz
,

and 1+Az

1+Bz
is the best dominant.

Putting g(z) = z(1− z)−1 and g(z) = z(1− z)−2, respectively, in Theorem 1, we have the

result obtained by Shanmugam et al. [24, Corollaries 3.2 and 3.3, respectively].

Taking g(z) of the form (5), and using the identity (see [9])

z
�

Hl ,s(α1) f (z)
�′
= α1Hl ,s(α1 + 1) f (z)− (α1− 1)Hl ,s(α1) f (z), (19)

then we have the following corollary.

Corollary 2. Let q be univalent in U and satisfies (10). If f ∈ S with Hl ,s(α1) f (z) 6= 0, z ∈ U∗,

and satisfies the subordination

χ1(α1,η,δ, f )≺ q(z) +
η

δ
zq′(z),

where χ1(α1,η,δ, f ) is given by

χ1(α1,η,δ, f ) = (1+α1η)

�
z

Hl ,s(α1) f (z)

�δ
−α1η

Hl ,s(α1+ 1) f (z)

Hl ,s(α1) f (z)

�
z

Hl ,s(α1) f (z)

�δ
, (20)

then �
z

Hl ,s(α1) f (z)

�δ
≺ q(z)

and q is the best dominant.

Letting g be of the form (6), and using the identity (see [7])

λz
�

Im(λ, l) f (z)
�′
= (l + 1)Im+1(λ, l) f (z)− (1+ l −λ)Im(λ, l) f (z) (λ > 0; l ¾ 0; m ∈ N0),

(21)

then we have the following corollary.

Corollary 3. Let q be univalent in U and satisfies (10), λ > 0, l ¾ 0 and m ∈ N0. If f ∈ S with

Im(λ, l) f (z) 6= 0, z ∈ U∗, and satisfies the subordination

χ2(l, m,λ,η,δ, f ) ≺ q(z) +
η

δ
zq′(z),



M. Aouf and A. Mostafa / Eur. J. Pure Appl. Math, 3 (2010), 641-652 647

where χ2(l, m,λ,η,δ, f ) is given by

χ2(l, m,λ,η,δ, f ) = (1+
η(l + 1)

λ
)
�

z

I m(λ,l) f (z)

�δ
−
η(l + 1)

λ

I m+1(λ,l) f (z)

I m(λ,l) f (z)

�
z

I m(λ,l) f (z)

�δ
, (22)

then �
z

Im(λ, l) f (z)

�δ
≺ q(z)

and q is the best dominant.

Theorem 2. Let γ ∈ C∗ and let q be univalent in U with q(0) = 1,q(z) 6= 0, z ∈ U and satisfies

the condition:

Re

¨
1+

zq′′(z)

q′(z)
−

zq′(z)

q(z)

«
> 0, z ∈ U . (23)

If f , g ∈ S with ( f ∗ g)(z) 6= 0, z ∈ U∗ and satisfies the subordination:

1+ γδ

�
1−

z( f ∗ g)′(z)

( f ∗ g)(z)

�
≺ 1+ γ

zq′(z)

q(z)
. (24)

then, �
z

( f ∗ g)(z)

�δ
≺ q(z),

and q is the best dominant of (24).

Proof. Let a function p defined by (14), then the function p is analytic in U and p(0) = 1.

Therefore, by differentiating (14) logarithmically with respect to z, we have

zp′(z)

p(z)
= δ

�
1−

z( f ∗ g)′(z)

( f ∗ g)(z)

�
.

Using the above relation in (24), we have

1+ γ
zp′(z)

p(z)
≺ 1+ γ

zq′(z)

q(z)
.

Taking θ(w) = 1 and ϕ(w) = γ/w, then ϕ and θ are analytic in C∗. Simple computations

show that

ψ(z) = zq′(z)ϕ(q(z)) = γ
zq′(z)

q(z)
,

h(z) = θ(q(z)) +ψ(z) = 1+ γ
zq′(z)

q(z)
,

and it is easily to see that the conditions of Lemma 1 are satisfied whenever (23) holds. Then,

applying Lemma 1, the proof of Theorem 2 is completed.

Putting q(z) = (1+ Az)/(1+ Bz) (−1≤ B < A≤ 1) in Theorem 2, it is easy to check that

the condition (23) holds whenever −1≤ B < A≤ 1, hence we obtain:



M. Aouf and A. Mostafa / Eur. J. Pure Appl. Math, 3 (2010), 641-652 648

Corollary 4. Let −1≤ B < A≤ 1 Let f , g ∈ S with ( f ∗ g)(z) 6= 0, z ∈ U∗, suppose that

1+ γδ

�
1−

z( f ∗ g)′(z)

( f ∗ g)(z)

�
≺ 1+

γ(A− B)z

(1+ Az)(1+ Bz)
.

Then, �
z

( f ∗ g)(z)

�δ
≺

1+ Az

1+ Bz
,

and (1+ Az)/(1+ Bz) is the best dominant.

Taking γ = −1

ab
(a, b ∈ C∗),δ = a and q(z) = (1− z)−2ab in Theorem 2, then combining

this together with Lemma 4, we obtain the following corollary.

Corollary 5. Let a, b ∈ C∗ such that |2ab− 1| ≤ 1 or |2ab+ 1| ≤ 1. Let f ∈ S and suppose that
( f ∗g)(z)

z
6= 0 for all z ∈ U∗. If

1+
1

b

�
z( f ∗ g)′(z)

( f ∗ g)(z)
− 1

�
≺

1+ z

1− z
,

then �
z

( f ∗ g)(z)

�a
≺ (1− z)−2ab,

and (1− z)−2ab is the best dominant.

Remark 1. (i) Taking g(z) = z

1−z
in Corollary 5, we obtain the result due to Obradovíc et al.

[17, Theorem 1];

(ii) Taking g(z) = z

1−z
and a = 1 in Corollary 5, we obtain the recent result of Srivastava and

Lashin [25, Theorem 3];

(iii) Taking g(z) = z

1−z
,γ = eiλ

ab cosλ
(a, b ∈ C∗; |λ| < π

2
), α = a and q(z) = (1− z)−2ab cosλe−iλ

in Corollary 5, we obtain the result due to Aouf et al. [3, Theorem 1].

Theorem 3. Let q be convex univalent in U, δ,η ∈ C∗ and satisfies

Re{
δ

η
} > 0. (25)

Let f , g ∈ S, ( f ∗ g)(z) 6= 0, z ∈ U∗, suppose that

�
z

( f ∗ g)(z)

�δ
∩ H[q(0), 1] ∈ Q and that

χg(α,η; f ) is univalent in U, where χg(δ,η; f ) is given by (12). Then

q(z) +
η

δ
zq′(z)≺ χg(δ,η; f )(z), (26)

implies

q(z) ≺
�

z

( f ∗ g)(z)

�δ
,

and q is the best subordinant of (26).



M. Aouf and A. Mostafa / Eur. J. Pure Appl. Math, 3 (2010), 641-652 649

Proof. Define a function p defined by (14). Then simple computations show that

p(z) +
η

δ
zp′(z) = χg(δ,η, f ).

Putting θ(w) = w and ϕ(w) = η/δ, then θ and ϕ are analytic in C , and

Re
θ ′(q(z))

ϕ(q(z))
= Re

δ

η
q′(z)> 0 (z ∈ U).

Since q is a convex function, it follows that h(z) = zq′(z)ϕ(q(z)) =
ηzq′(z)

δ
is starlike in U .

Then by applying Lemma 3, the proof is completed.

Letting g be of the form (5) in Theorem 3 and using the identity (19), we get the following

result obtained the following result:

Corollary 6. Let q be convex in U, and suppose that δ,η ∈ C∗ satisfies the condition (25). For all

functions f ∈ S with Hl ,s(α1) f (z) 6= 0, z ∈ U∗, suppose that

�
z

Hl ,s(α1) f (z)

�α
∈ H[q(0), 1]∩Q,

and that χ1(α1;δ,η; f ) is univalent in U, where χ1(α1;δ,η; f ) is given by (20).

Then,

q(z) +
η

δ
zq′(z) ≺ χ1(α1;δ,η; f )(z), (27)

implies

q(z) ≺

�
z

Hl ,s(α1) f (z)

�δ
,

and q is the best subordinant of (27).

Letting g be of the form (6) in Theorem 3 and using the identity (21), we have:

Corollary 7. Let q be convex in U, and suppose that α,η ∈ C∗ satisfies the condition (25).

For all functions f ∈ S with I(m,λ, l) f (z) 6= 0, z ∈ U∗
�
λ > 0, l ≥ 0, m ∈ N0

�
, suppose

that

�
z

I(m,λ, l) f (z)

�δ
∈ H[q(0), 1]∩Q, and that χ2(m,λ, l;δ,η; f ) is univalent in U, where

χ2(m,λ, l;δ,η; f ) is given by (22).

Then,

q(z) +
η

α
zq′(z) ≺ χ2(m,λ, l;α,η; f )(z), (28)

implies

q(z)≺
�

z

I(m,λ, l) f (z)

�α
,

and q is the best subordinant of (28).

Combining Theorem 1 and Theorem 3, we deduce the following sandwich theorem:



M. Aouf and A. Mostafa / Eur. J. Pure Appl. Math, 3 (2010), 641-652 650

Theorem 4. Let q1 and q2 be convex functions in U. Suppose that δ,η ∈ C∗ satisfies (25) and

q2 satisfies (10). Let f , g ∈ S , with ( f ∗ g)(z) 6= 0, z ∈ U∗, suppose that

�
z

( f ∗ g)(z)

�δ
∈

H[q(0), 1]∩Q, and that χg(δ,η; f ) is univalent in U, where χg(δ,η; f ) is given by (12). Then,

q1(z) +
η

δ
zq′1(z) ≺ χg(δ,η; f )(z)≺ q2(z) +

η

δ
zq′2(z), (29)

implies

q1(z) ≺
�

z

( f ∗ g)(z)

�δ
≺ q2(z),

and q1 and q2 are respectively, the best subordinant and the best dominant.

Combining Corollary 2 and Corollary 6, we get the sandwich result:

Corollary 8. Let q1 and q2 be convex functions in U. Suppose that δ,η ∈ C∗ satisfies (25) and

q2 satisfies (10). Let f ∈ S , with Hl ,s(α1) f (z) 6= 0, z ∈ U∗, suppose that

�
z

Hl ,s(α1) f (z)

�δ
∈

H[q(0), 1]∩Q, and that χ1(α1;δ,η; f ) is univalent in U, where χ1(α1;δ,η; f ) is given by (20).

Then,

q1(z) +
η

δ
zq′1(z) ≺ χ1(α1;δ,η; f )≺ q2(z) +

η

δ
zq′2(z),

implies

q1(z) ≺

�
z

Hl ,s(α1) f (z)

�δ
≺ q2(z),

and q1 and q2 are respectively, the best subordinant and the best dominant.

Combining Corollary 3 and Corollary 7, we get the sandwich result:

Corollary 9. Let q1 and q2 be convex functions in U. Suppose that δ,η ∈ C∗ satisfies (25) and

q2 satisfies (10). Let f ∈ S , with I(m,λ, l) f (z) 6= 0, z ∈ U∗, suppose that

�
z

I(m,λ, l) f (z)

�δ
∈

H[q(0), 1]∩Q, and that χ2(m,λ, l;α,η; f ) is univalent in U, where χ2(m,λ, l;α,η; f ) is given

by (22). Then,

q1(z) +
η

δ
zq′1(z) ≺ χ2(m,λ, l;α,η; f )(z)≺ q2(z) +

η

δ
zq′2(z),

implies

q1(z) ≺
�

z

I(m,λ, l) f (z)

�δ
≺ q2(z),

and q1 and q2 are respectively, the best subordinant and the best dominant.

Remark 2. Taking g in the form (4) in Theorems 1, 3 and 4, respectively, we obtain the results

obtained by Shanmugam et al. [ 24, Theorems, 3.1, 4.1 and 5.1, respectively].

Specializing the parameters α j( j = 1,2, ..., s + 1), β j( j = 1,2, ..., s) , λ, l and m, in Corol-

laries 8 and 9, we obtain the sandwich results for the corresponding operators.



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