ISRN Mathematical Analysis

Volume 2014 (2014), Article ID 606235, 5 pages

http://dx.doi.org/10.1155/2014/606235

Research Article

Some Applications of Second-Order Differential Subordination on a Class of Analytic Functions Defined by Komatu Integral Operator

Serap Bulut

Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 Kartepe-Kocaeli, Turkey

Received 24 December 2013; Accepted 13 February 2014; Published 12 March 2014

Academic Editors: G. Mantica, C. Mascia, A. Peris, and W. Shen

Copyright © 2014 Serap Bulut. 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.

Abstract.

We introduce a new class of analytic functions by using Komatu integral operator and obtain some subordination results.

1. Introduction, Definitions, and Preliminaries

Let be the set of real numbers, the set of complex numbers,

be the set of positive integers, and

Let be the class of analytic functions in the open unit disk

and the subclass of consisting of the functions of the form

Let be the class of all functions of the form

which are analytic in the open unit disk with

Also let denote the subclass of consisting of functions which are univalent in .

A function analytic in is said to be convex if it is univalent and is convex.

Let

denote the class of normalized convex functions in .

If and are analytic in , then we say that is subordinate to , written symbolically as

if there exists a Schwarz function which is analytic in with

such that

Indeed, it is known that

Furthermore, if the function is univalent in , then we have the following equivalence [1, page 4]:

Let be a function and let be univalent in . If is analytic in and satisfies the (second-order) differential subordination

then is called a solution of the differential subordination. The univalent function is called a dominant of the solutions of the differential subordination, or more simply a dominant, if for all satisfying (13).

A dominant , which satisfies for all dominants of (13), is said to be the best dominant of (13).

Recently, Komatu [2] introduced a certain integral operator defined by

Thus, if is of the form (5), then it is easily seen from (14) that (see [2])

Using the relation (15), it is easy verify that

We note the following.

(i)For and ( is any integer), the multiplier transformation was studied by Flett [3] and Sălăgean [4].(ii)For and (), the differential operator was studied by Sălăgean [4].(iii)For and ( is any integer), the operator was studied by Uralegaddi and Somanatha [5].(iv)For , the multiplier transformation was studied by Jung et al. [6].

Using the operator , we now introduce the following class.

Definition 1. Let be the class of functions satisfying

where , , and is the Komatu integral operator.

In order to prove our main results, we will make use of the following lemmas.

Lemma 2 (see [7]). Let be a convex function with and let be a complex number with . If and

then

where

The function is convex and is the best dominant.

Lemma 3 (see [8]). Let , , and let

Let be an analytic function in with and suppose that

If

is analytic in and

then

where is a solution of the differential equation

given by

Moreover is the best dominant.

2. Main Results

Theorem 4. The set is convex.

Proof. Let

be in the class . Then, by Definition 1, we have

For any nonnegative numbers such that

we must show that the function

is in ; that is,

By (28) and (31), we have

Therefore we get

Differentiating (34) with respect to , we obtain

So we get

since . Therefore we get the desired result.

Theorem 5. Let be convex function in with and let

where is a complex number with . If and , where

then

implies

and this result is sharp.

Proof. From the equality (38), we get

Differentiating (41) with respect to , we have

Differentiating (43) with respect to , we obtain

Using the differential subordination (39) in the equality (44), we get

Let us define

Then a simple computation yields

Using (46) in the subordination (45), we have

Using Lemma 2, we obtain

which is desired result. Moreover is the best dominant.

Example 6. If we take

in Theorem 5, then we have

If and is given by

then by Theorem 5, we have

Theorem 7. Let and let

Let be an analytic function in with and suppose that

If and , where is defined by (38), then

implies

where is the solution of the differential equation

given by

Moreover is the best dominant.

Proof. We consider and in Lemma 3. Then the proof is easily seen by means of the proof of Theorem 5.

Letting

in Theorem 7, we obtain the following interesting result.

Corollary 8. If , , , , and is defined by (38), then

where

and this results is sharp. Moreover

where

Proof. If we let

then is convex and by Theorem 7, we deduce

On the other hand if , then from the convexity of and the fact that is symmetric with respect to the real axis, we get

where is given by (64). From (66), we have

where is given by (63).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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