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

**Academic Editor:**W. Shen

#### 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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#### Copyright

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.