ABSTRACT

In this investigation, we obtain some applications of first order differential subordination and superordination results involving an extended multiplier transformation and other linear operators for certain normalized analytic functions. Some of our results improve previous results.

1. Introduction

Let be the class of functions analytic in the open unit disk. Let be the subclass of consisting of functions of the form:

(1.1)

For simplicity, let. Also, let be the subclass of consisting of functions of the form:

(1.2)

If we say that is subordinate to written if there exists Schwarz function which (by definition) is analytic in with and for all such that Furthermore, if the function is univalent in then we have the following equivalence , (cf., e.g. [1,2]; see also [3]): 

We denote this subordination by

and

Let and let. If p and are univalent and if p satisfies the second-order superordination

, (1.3)

then is a solution of the differential superordination (1.3). Note that if is subordinate to, then is superordinate to. An analytic function is called a subordinant if for all satisfying (1.3). A univalent subordinant that satisfies for all subordinants of (1.3) is called the best subordinant. Recently Miller and Mocanu [4] obtained conditions on the functions and for which the following implication holds:

(1.4)

Using the results of Miller and Mocanu [4], Bulboaca [5] considered certain classes of first-order differential superordinations as well as superordination-preserving integral operators [6]. Ali et al. [7] have used the results of Bulboaca [5] and obtained sufficient conditions for normalized analytic functions to satisfy:

where and are given univalent functions in. Also, Tuneski [8] obtained a sufficient condition for starlikeness of in terms of the quantity.

Recently Shanmugam et al. [9] obtained sufficient conditions for a normalized analytic functions to satisfy

and

Many essentially equivalent definitions of multiplier transformation have been given in literature (see [10-12]. In [13] Catas defined the operator as follows:

Definition 1.1. [13] Let the function. For where The extended multiplier transformation on is defined by the following infinite series:

(1.5)

It follows form (1.5) that

(1.6)

and

(1.7)

for all integers and. We note that:

1) (see [14]);

2) (see [15]);

3) (see [10,11]);

4) (see [12]).

Also if, then we can write

where

In this paper, we obtain sufficient conditions for the normalized analytic function defined by using an extended multiplier transformation to satisfy:

and

and and are given univalent functions in.

2. Definitions and Preliminaries

In order to prove our results, we shall make use of the following known results.

Definition 2.1. [4]

Denote by the set of all functions that are analytic and injective on where

and are such that for

Lemma 2.1. [4]

Let the function be univalent in the open unit disc and and be analytic in a domain containing with when. Set

. (2.1)

Suppose that 1) is starlike univalent in2) for.

If is analytic with and

(2.2)

then and is the best dominant. Taking and in lemma 1, Shanmugam et al. [9] obtained the following lemma.

Lemma 2.2. [2]

Let be univalent in with Let ; further assume that

If is analytic in, and

then and is the best dominant.

Lemma 2.3. [5]

Let the function be univalent in the open unit disc and and be analytic in a domain containing Suppose that 1) for and 2) is starlike univalent in.

If with,

, is univalent in and

(2.3)

then and is the best subordinant.

Taking and in Lemma 2.3, Shanmugam et al. [9] obtained the following lemma.

Lemma 2.4. [2]

Let be convex univalent in, Let, and If is univalent in and then and is the best subordinant.

3. Applications to an Extended Multiplier Transformation and Sandwich Theorems

Theorem 3.1.

Let be convex univalent in with Further, assume that

(3.1)

If, for and

(3.2)

then

and is the best dominant.

Proof. Define a function by

(3.3)

Then the function is analytic in and. Therefore, differentiating (3.3) logarithmically with respect to and using the identity (1.6) in the resulting equation, we have

that is,

and therefore, the theorem follows by applying Lemma 2.2.

Putting

in Theorem 3.1, we have the following corollary.

Corollary 3.1.

If and satisfy

then

Putting and in Corollary 3.1, we have

Corollary 3.2.

If and satisfy

then

Taking in Theorem 1, we have

Corollary 3.3.

Let be convex univalent in with . Further, assume that (3.1) holds. If, and

then

and is the best dominant.

Taking in Theorem 3.1, we have

Corollary 3.4. 

Let be convex univalent in with . Further, assume that (3.1) holds. If, and

then

and is the best dominant.

Taking in Theorem 3.1, we have

Corollary 3.5.

Let be convex univalent in with . Further, assume that (3.1) holds. If, and

then

and is the best dominant.

Taking in Theorem 1, we have

Corollary 3.6. 

Let be convex univalent in with . Further, assume that (3.1) holds. If, and

then

and is the best dominant.

Now, by appealing to Lemma 2.4 it can be easily prove the following theorem.

Theorem 3.2.

Let be convex univalent in. Let with

If,

is univalent in, and

then

and is the best subordinant.

Taking, in Theorem 3.2, we have

Corollary 3.7. 

Let be convex univalent in. Let with

If,

is univalent in, and

then

and is the best subordinant.

Taking in Theorem 3.2, we have

Corollary 3.8. 

Let be convex univalent in. Let with

If,

is univalent in, and

then

and is the best subordinant.

Taking in Theorem 3.2, we have

Corollary 3.9.

Let be convex univalent in. Let with

If,

is univalent in, and

then

and is the best subordinant.

Taking in Theorem 3.2, we have

Corollary 3.10. 

Let be convex univalent in. Let with

If,

is univalent in, and

then

and is the best subordinant.

Combining Theorems 3.1 and 3.2, we get the following sandwich theorem.

Theorem 3.3.

Let be convex univalent in, with be univalent in and satisfies (3.1). If

is univalent in, and

Then

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

4. Remarks

Combining: 1) Corollary 3.3 and Corollary 3.7; 2) Corollary 3.4 and Corollary 3.8; 3) Corollary 3.5 and Corollary 3.9; 4) Corollary 3.6 and Corollary 3.10, we obtain similar sandwich theorems for the corresponding operators.

Theorem 3.4.

Let be convex univalent in,. Further, assume that (3.1) holds.

If satisfies

then

and q is the best dominant.

Proof. Define the function by

.

Then, simple computations show that

Applying Lemma 2, the theorem follows.

Taking in Theorem 3.4, we have the following corollary.

Corollary 3.11.

Let be convex univalent in,. Further, assume that (3.1) holds. If satisfies

then

and is the best dominant.

Taking in Theorem 3.4, we have  

Corollary 3.12.

Let be convex univalent in,. Further, assume that (3.1) holds. If satisfies

then

and is the best dominant.

Taking in Theorem 3.4, we have

Corollary 3.13.

Let be convex univalent in,. Further, assume that (3.1) holds. If satisfies

then

and is the best dominant.

Taking in Theorem 3.4, we have

Corollary 3.14.

Let be convex univalent in,. Further, assume that (3.1) holds. If satisfies

then

and is the best dominant.

Theorem 3.5. 

Let be convex univalent in. Let with

If,

is univalent in and

then

and is the best subordinant.

Proof. The proof follows by applying Lemma 3.4.

Combining Theorems 3.4 and 3.5, we get the following sandwich theorem.

Theorem 3.6.

Let be convex univalent in, with be univalent in and satisfies (3.1). If,

is univalent in and

then

and and are respectively the best subordinant and the best dominant.

REFERENCES