Some Subordination Results on 𝑞-Analogue of Ruscheweyh Differential Operator

Abstract

We derive the -analogue of the well-known Ruscheweyh differential operator using the concept of -derivative. Here, we investigate several interesting properties of this -operator by making use of the method of differential subordination.

1. Introduction

Recently, the area of -analysis has attracted the serious attention of researchers. This great interest is due to its application in various branches of mathematics and physics. The application of -calculus was initiated by Jackson [1, 2]. He was the first to develop -integral and -derivative in a systematic way. Later, from the 80s, geometrical interpretation of -analysis has been recognized through studies on quantum groups. It also suggests a relation between integrable systems and -analysis. In ([3–5]) the -analogue of Baskakov Durrmeyer operator has been proposed, which is based on -analogue of beta function. Another important -generalization of complex operators is -Picard and -Gauss-Weierstrass singular integral operators discussed in [6–8]. The authors studied approximation and geometric properties of these -operators in some subclasses of analytic functions in compact disk. Very recently, other -analogues of differential operators have been introduced in [9]; see also ([10, 11]). These -operators are defined by using convolution of normalized analytic functions and -hypergeometric functions, where several interesting results are obtained. From this point, it is expected that deriving -analogues of operators defined on the space of analytic functions would be important in future. A comprehensive study on applications of -analysis in operator theory may be found in [12].

We provide some notations and concepts of -calculus used in this paper. All the results can be found in [12–14]. For , we define

As , and this is the bookmark of a -analogue: the limit as recovers the classical object.

For complex parameters , the -analogue of Gauss’s hypergeometric function is defined by where is the -analogue of Pochhammer symbol defined by

The -derivative of a function is defined by and . For a function observe that then , where is the ordinary derivative.

Next, we state the class of all functions of the following form: which are analytic in the open unit disk . If and are analytic functions in , we say that is subordinate to ; written , if there is a function analytic in , with , for all , such that for all . If is univalent, then if and only if and .

For each and such that , we define the function It is well known that for is the conformal map of the unit disk onto the disk symmetrical with respect to the real axis having the center for and radius . The boundary circle cuts the real axis at the points and .

Definition 1. Let . Denote by the -analogue of Ruscheweyh operator defined by where and are defined in (1).

From the definition we observe that, if , we have where is Ruscheweyh differential operator which was defined in [15] and has been studied by many authors, for example [16–18].

It can also be shown that this -operator is hypergeometric in nature as where is the -analogue of Gauss hypergeometric function defined in (2), and the symbol stands for the Hadamard product (or convolution).

The following identity is easily verified for the operator :

2. Main Results

Before we obtain our results, we state some known lemmas.

Let be the class of functions of the form which are analytic in and satisfy the following inequality:

Lemma 2 (see [19]). Let be given by (12), where ; then and the bound is the best possible.

Lemma 3 (see [20]). Let the function , given by (12), be in the class . Then

Lemma 4 (see [21]). The function , is univalent in if and only if is either in the closed disk or in the closed disk .

We now generalize the lemmas introduced in [22] and [23], respectively, using -derivative.

Lemma 5. Let be analytic and convex univalent in and and let be analytic in . If Then, for ,

Proof. Suppose that is analytic and convex univalent in and is analytic in . Letting in (16), we have
Then, from Lemma in [22], we obtain

Lemma 6. Let be univalent in and let and be analytic in a domain containing with when . Set and suppose that (1) is starlike univalent in ;(2).
If is analytic in , with , and then and is the best dominant.

The proof is similar to the proof of Lemma 5.

Theorem 7. Let , , and . If satisfies then
The result is sharp.

Proof. Let for . Then the function is analytic in . By using logarithmic -differentiation on both sides of (23) and multiplying by , we have by making use of identity (11), we obtain
Taking into account that , we obtain
From (11), (23), and (26), we get
Now, applying Lemma 5, we have or by the concept of subordination
In view of and , it follows from (29) that with the aid of the elementary inequality for and . Hence, inequality (22) follows directly from (30). To show the sharpness of (22), we define by
For this function, we find that
This completes the proof.

Corollary 8. Let , where and . If satisfies then

Proof. Following the same steps as in the proof of Theorem 7 and considering , the differential subordination (27) becomes
Therefore,

Theorem 9. Let and . Let be a complex number with and satisfy either or . If satisfies the condition then where is the best dominant.

Proof. Let Then, by making use of (11), (37), and (39), we obtain
We now assume that then is univalent by condition of the theorem and Lemma 4. Further, it is easy to show that , and satisfy the conditions of Lemma 6. Note that the function is univalent starlike in and
Combining (40) and Lemma 6 we get the assertion of Theorem 9.

Theorem 10. Let and . If each of the functions satisfies the following subordination condition, then, where

Proof. we define the function by we have , where . By making use of (11) and (48), we obtain which, in the light of (46), can show that where, for convenience,
Note that, by using Lemma 2, we have , where .
Now, with an application of Lemma 3, we have which shows that the desired assertion of Theorem 10 holds.

Conflict of Interests

The authors declare that they have no competing interests regarding the publication of this paper.

Authors’ Contribution

Huda Aldweby and Maslina Darus read and approved the final manuscript.

Acknowledgments

The work presented here was partially supported by AP-2013-009 and DIP-2013-001. The authors also would like to thank the referees for the comments made to improve this paper.