Differential Subordination with Generalized Derivative Operator of Analytic Functions

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.