Abstract

We investigate convolution properties and coefficients estimates for two classes of analytic functions involving the -derivative operator defined in the open unit disc. Some of our results improve previously known results.

1. Introduction

Simply, -calculus or -calculus is ordinary classical calculus without the notion of limits. Here ostensibly stands for Planck’s constant, while stands for quantum. Recently, the area of -calculus 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, geometrical interpretation of -analysis has been recognized through studies on quantum groups. It also suggests a relation between integrable systems and -analysis. Aral and Gupta [3–5] defined and studied the -analogue of Baskakov Durrmeyer operator 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]. Mohammed and Darus [9] studied approximation and geometric properties of these -operators in some subclasses of analytic functions in compact disk. These -operators are defined by using convolution of normalized analytic functions and -hypergeometric functions, where several interesting results are obtained (see also [10, 11]). A comprehensive study on applications of -calculus in operator theory may be found in [12].

Let denote the class of functions of the form: which are analytic in the open unit disk . Let and denote the subclasses of that consists, respectively, of starlike of order and convex of order in (see [13]). If and are analytic in , we say that is subordinate to , written if there exists a Schwarz function , which (by definition) is analytic in with and for all , such that , . Furthermore, if the function is univalent in , then we have the following equivalence (see [14–16]):

For functions given by (1) and given by the Hadamard product or convolution of and is defined by Let and denote the subclasses of the class for which are defined by (see [17–22]) We note that For function given by (1) and , the -derivative of a function is defined by (see [1]) and . From (7), we deduce that where As , . For a function , we observe that where is the ordinary derivative.

Making use of the -derivative , we introduce the subclasses and of for and as follows: We note that(i)(ii)(iii)(iv)(v)

From (11), we have In this paper, we investigate convolution properties, the necessary and sufficient condition and coefficient estimates for the classes and associated with the -derivative . The motivation of this paper is to improve and generalize previously known results.

2. Convolution Properties

Unless otherwise mentioned, we assume throughout this section that , and .

Theorem 1. The function defined by (1) is in the class if and only if for all and also .

Proof. First suppose defined by (1) is in the class ; we have Since the function from the left-hand side of the subordination is analytic in , it follows , ; that is, , and this is equivalent to the fact that (18) holds for . From (19) according to the subordination of two analytic functions we say that there exists a function analytic in with , such that which is equivalent to or Since Now from (23), we may write (22) as which leads to (18), which proves the necessary part of Theorem 1.
Reversely, because assumption (18) holds for , it follows that for all ; hence, the function is analytic in (i.e., it is regular at , with ). Since it was shown in the first part of the proof that assumption (18) is equivalent to (21), we obtain that and if we denote relation (25) shows that . Thus, the simply connected domain is included in a connected component of . From here, using the fact that together with the univalence of the function , it follows that , which represents in fact subordination (19); that is, . This completes the proof of Theorem 1.

Taking in Theorem 1, we obtain the following result which improves the convolution result of Aouf and Seoudy [23, Theorem 1] and also the result of Silverman and Silvia [21, Theorem 7].

Corollary 2. The function defined by (1) is in the class if and only if for all and also .

Putting and in Theorem 1, we obtain the following corollary.

Corollary 3. The function defined by (1) is in the class if and only if for all , and also .

Taking in Corollary 3, we obtain the following result which improves the convolution result of Silverman et al. [22, Theorems 1].

Corollary 4. The function defined by (1) is in the class if and only if for all , , and also .

Theorem 5. The function defined by (1) is in the class if and only if for all and also .

Proof. Set and we note that From the identity and the fact that the result follows from Theorem 1.

Taking in Theorem 1, we obtain the following result which improves the result of Aouf and Seoudy [23, Theorem 2].

Corollary 6. The function defined by (1) is in the class if and only if for all and also .

Putting and in Theorem 5, we obtain the following corollary.

Corollary 7. The function defined by (1) is in the class if and only if for all , , and also .

Taking in Corollary 7, we obtain the following result which improves the convolution result of Silverman et al. [22, Theorem 2].

Corollary 8. The function defined by (1) is in the class if and only if for all , , and also .

Theorem 9. A necessary and sufficient condition for the function defined by (1) to be in the class is that

Proof. From Theorem 1, we find that if and only if for all and also for . The left-hand side of (38) can be written as Thus, the proof of The Theorem 9 is completed.

Taking in Theorem 9, we obtain the following result.

Corollary 10. A necessary and sufficient condition for the function defined by (1) to be in the class is that

Putting and in Theorem 9, we obtain the following corollary.

Corollary 11. A necessary and sufficient condition for the function defined by (1) to be in the class is that

Taking in Corollary 11, we obtain the following corollary which improves the result of Ahuja [17, Corollary 1 when ].

Corollary 12. A necessary and sufficient condition for the function defined by (1) to be in the class is that

Theorem 13. A necessary and sufficient condition for the function defined by (1) to be in the class is that

Proof. From Theorem 5, we find that if and only if for all and also for . The left-hand side of (44) may be written as and this proves Theorem 13.

Taking in Theorem 13, we obtain the following result.

Corollary 14. A necessary and sufficient condition for the function defined by (1) to be in the class is that

Putting and in Theorem 13, we obtain the following corollary.

Corollary 15. A necessary and sufficient condition for the function defined by (1) to be in the class is that

Taking in Corollary 15, we obtain the following corollary which improves the result of Ahuja [17, Corollary 1 when ].

Corollary 16. A necessary and sufficient condition for the function defined by (1) to be in the class is that

3. Coefficient Estimates

As an application of Theorems 9 and 13, we next determine coefficient estimate and inclusion property for a function of form (1) to be in the classes and .

Theorem 17. If the function defined by (1) satisfies the following inequality: then .