Research Article | Open Access

# Differential Subordination with Generalized Derivative Operator of Analytic Functions

**Academic Editor:**Annamaria Barbagallo

#### Abstract

Motivated by generalized derivative operator defined by the authors (El-Yagubi and Darus, 2013) and the technique of differential subordination, several interesting properties of the operator are given.

#### 1. Introduction

Let denote the class of functions of the form which are analytic in the open unit disk .

Also let be the the subclass of consisting of all functions which are univalent in . We denote by and the familiar subclasses of consisting of functions which are, respectively, starlike of order and convex of order in :.

Let be the class of holomorphic function in unit disk . For and we let

Let two functions given by and be analytic in . Then the Hadamard product (or convolution) of the two functions , is defined by

Recall that the function is subordinate to if there exists the Schwarz function , analytic in , with and such that , . We denote this subordination by . If is univalent in , then the subordination is equivalent to and .

Let and 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 (5).

A dominant that satisfies for all dominants of (5) is said to be the best dominant of (5) (note that the best dominant is unique up to a rotation of ).

In order to prove the original results we need the following lemmas.

Lemma 1 (see [1]). *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 2 (see [2]). *Let be a convex function in and let
**
where and is a positive integer. If
**
is analytic in and
**
then
**
and this result is sharp.*

Lemma 3 (see [3]). *Let ; if
**
then
**
belongs to the class of convex functions.*

We now state the following generalized derivative operator [4]: where , for , and is the Pochhammer symbol defined by

Here can also be written in terms of convolution as

To prove our results, we need the following inclusion relation: where is analytic function given by .

#### 2. Main Results

In the present paper, we will use the method of differential subordination to derive certain properties of generalised derivative operator . Note that differential subordination has been studied by various authors, and here we follow similar works done by Oros [5] and G. Oros and G. I. Oros [6].

*Definition 4. *For , and , let denote the class of functions which satisfy the condition

Also, let denote the class of functions which satisfy the condition

*Remark 5. *It is clear that , and the class of functions satisfy
studied by Ponnusamy [7] and others.

Theorem 6. *Let
**
be convex in , with and .**If , and and satisfies the differential subordination
**
then
**
where is given by
**
The function is convex and is the best dominant.*

* Proof. *By differentiating (18), with respect to , we obtain

Using (26) in (23), the differential subordination (23) becomes
Let

Using (28) in (27), the differential subordination becomes

By using Lemma 1, we have
where is given by (25); that is,
The function is convex and is the best dominant. The proof is complete.

Theorem 7. *If , and , then one has
**
where
**
and is given by (25).*

* Proof. *Let , and then from (19) we have
which is equivalent to

Using Theorem 6, we have

Since is convex and is symmetric with respect to the real axis, we deduce that
for which we deduce . The proof is complete.

Theorem 8. *Let be a convex function in , with , and let
**
If , and satisfies the differential subordination
**
then
**
and the result is sharp.*

* Proof. *Using (28) in (26), the differential subordination (39) becomes

Using Lemma 2, we have
that is,
and the result is sharp. The proof of Theorem 8 is complete.

*Example 9. *For , and , by applying Theorem 8, we have

By using equality (18) we find that

Now,

A straightforward calculation gives the following:

Similarly, using (18), we see that
and then

By using (47) we have
we deduce
that is,

From Theorem 8 we get
which implies that

Theorem 10. *Let be a convex function in , with , and let
**
If , and satisfies the differential subordination
**
then
**
and the result is sharp.*

* Proof. *Let

Differentiating (58), with respect to , we obtain

Using (58), the differential subordination (56) becomes

Using Lemma 2, we deduce that

By using (58), we have
The proof of Theorem 10 is complete.

*Example 11. *For , and , from Theorem 10 we obtain
From Example 9, we have
and then
From Theorem 10 we deduce that
implies that

Theorem 12. *Let be a convex function in , with , and let
**
If , and satisfies the differential subordination
**
then
**
The function is convex and is the best dominant.*

* Proof. *Let

Differentiating (71), with respect to , we obtain

Using (72), the differential subordination (69) becomes

Using Lemma 1, we deduce that

By using (71), we have
The proof of Theorem 12 is complete.

Corollary 13. *If , then
*

*Proof. *Since , from Definition 4 we have
which is equivalent to

Using Theorem 12, we obtain

Since is convex and is symmetric with respect to the real axis, we have that

Theorem 14. *Let , with , , which satisfies the inequality
**
If , and satisfies the differential subordination
**
then
*

*Proof. *Let

Differentiating (84), with respect to , we obtain

Using (85), the differential subordination (82) becomes

Using Lemma 1, we deduce that
by using (84), we have
From Lemma 3, we see that the function is convex, and from Lemma 1, is the best dominant for subordination (82). The proof of Theorem 14 is complete.

Note that other work related to differential operators and differential subordination can be seen in [8–13].

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Authors’ Contribution

Entisar El-Yagubi and Maslina Darus read and approved the final paper.

#### Acknowledgment

The work presented here was partially supported by LRGS/TD/2013/UKM/ICT/03/02.

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#### Copyright

Copyright © 2014 Entisar El-Yagubi and Maslina Darus. 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.