Some Properties of Certain Class of Integral Operators

Abstract

The main object of this paper is to derive some inequality properties and convolution properties of certain class of integral operators defined on the space of meromorphic functions.

1. Introduction and Preliminaries

Let denote the class of functions of the form

(1.1)

which are analytic in the punctured open unit disk

(1.2)

Let , where is given by (1.1) and is defined by

(1.3)

Then the Hadamard product (or convolution) of the functions and is defined by

(1.4)

For two functions and , analytic in , we say that the function is subordinate to in and write

(1.5)

if there exists a Schwarz function , which is analytic in with

(1.6)

such that

(1.7)

Indeed, it is known that

(1.8)

Furthermore, if the function is univalent in , then we have the following equivalence:

(1.9)

Analogous to the integral operator defined by Jung et al. [1], Lashin [2] recently introduced and investigated the integral operator

(1.10)

defined, in terms of the familiar Gamma function, by

(1.11)

By setting

(1.12)

we define a new function in terms of the Hadamard product (or convolution)

(1.13)

Then, motivated essentially by the operator , Wang et al. [3] introduced the operator

(1.14)

which is defined as

(1.15)

where (and throughout this paper unless otherwise mentioned) the parameters , and are constrained as follows:

(1.16)

and is the Pochhammer symbol defined by

(1.17)

Clearly, we know that .

It is readily verified from (1.15) that

(1.18)

(1.19)

Recently, Wang et al. [3] obtained several inclusion relationships and integral-preserving properties associated with some subclasses involving the operator , some subordination and superordination results involving the operator are also derived. Furthermore, Sun et al. [4] investigated several other subordination and superordination results for the operator .

In order to derive our main results, we need the following lemmas.

Lemma 1.1 (see [5]).

Let be analytic and convex univalent in with . Suppose also that is analytic in with . If

(1.20)

then

(1.21)

and is the best dominant of (1.20).

Let denote the class of functions of the form

(1.22)

which are analytic in and satisfy the condition

(1.23)

Lemma 1.2 (see [6]).

Let

(1.24)

Then

(1.25)

The result is the best possible.

Lemma 1.3 (see [7]).

Let

(1.26)

Then

(1.27)

In the present paper, we aim at proving some inequality properties and convolution properties of the integral operator .

2. Main Results

Our first main result is given by Theorem 2.1 below.

Theorem 2.1.

Let and . If satisfies the condition

(2.1)

then

(2.2)

The result is sharp.

Proof.

Suppose that

(2.3)

Then is analytic in with . Combining (1.18) and (2.3), we find that

(2.4)

From (2.1), (2.3), and (2.4), we get

(2.5)

By Lemma 1.1, we obtain

(2.6)

or equivalently,

(2.7)

where is analytic in with

(2.8)

Since and , we deduce from (2.7) that

(2.9)

By noting that

(2.10)

the assertion (2.2) of Theorem 2.1 follows immediately from (2.9) and (2.10).

To show the sharpness of (2.2), we consider the function defined by

(2.11)

For the function defined by (2.11), we easily find that

(2.12)

it follows from (2.12) that

(2.13)

This evidently completes the proof of Theorem 2.1.

In view of (1.19), by similarly applying the method of proof of Theorem 2.1, we get the following result.

Corollary 2.2.

Let and . If satisfies the condition

(2.14)

then

(2.15)

The result is sharp.

For the function given by (1.1), we here recall the integral operator

(2.16)

defined by

(2.17)

Theorem 2.3.

Let , and . Suppose also that is given by (2.17). If satisfies the condition

(2.18)

then

(2.19)

The result is sharp.

Proof.

We easily find from (2.17) that

(2.20)

Suppose that

(2.21)

It follows from (2.18), (2.20) and (2.21) that

(2.22)

The remainder of the proof of Theorem 2.3 is much akin to that of Theorem 2.1, we therefore choose to omit the analogous details involved.

Theorem 2.4.

Let and . If is defined by

(2.23)

and each of the functions satisfies the condition

(2.24)

then

(2.25)

The result is sharp when .

Proof.

Suppose that satisfy conditions (2.24). By setting

(2.26)

it follows from (2.24) and (2.26) that

(2.27)

Combining (1.18) and (2.26), we get

(2.28)

For the function given by (2.23), we find from (2.28) that

(2.29)

where

(2.30)

By noting that and , it follows from Lemma 1.2 that

(2.31)

Furthermore, by Lemma 1.3, we know that

(2.32)

In view of (2.24), (2.30), and (2.32), we deduce that

(2.33)

When , we consider the functions which satisfy conditions (2.24) and are given by

(2.34)

It follows from (2.26), (2.28), (2.30), and (2.34) that

(2.35)

Thus, we have

(2.36)

The proof of Theorem 2.4 is evidently completed.

With the aid of (1.19), by applying the similar method of the proof of Theorem 2.4, we obtain the following result.

Corollary 2.5.

Let and . If is defined by (2.23) and each of the functions satisfies the condition

(2.37)

then

(2.38)

The result is sharp when .