Convolution Properties for Certain Classes of Analytic Functions Defined by q-Derivative Operator

and Applied Analysis 3 2. Convolution Properties Unless otherwise mentioned, we assume throughout this section that θ ∈ [0, 2π), 0 < q < 1 and −1 ≤ B < A ≤ 1. Theorem 1. The function f defined by (1) is in the class S q [A, B] if and only if 1 z [f (z) ∗ z − Lqz 2 (1 − z) (1 − qz) ] ̸ = 0 (z ∈ U) (18) for all L = L θ = (e + A)/(A − B) and also L = 1. Proof. First suppose f defined by (1) is in the class S q [A, B]; we have zD q f (z) f (z) ≺ 1 + Az 1 + Bz . (19) Since the function from the left-hand side of the subordination is analytic inU, it followsf(z) ̸ = 0, z ∈ U∗ = U\{0}; that is, (1/z)f(z) ̸ = 0, z ∈ U, and this is equivalent to the fact that (18) holds forL = 1. From (19) according to the subordination of two analytic functions we say that there exists a function w(z) analytic in U with w(0) = 0, |w(z)| < 1 such that zD q f (z) f (z) = 1 + Aw (z) 1 + Bw (z) (z ∈ U) (20) which is equivalent to zD q f (z) f (z) ̸ = 1 + Ae iθ 1 + Be (z ∈ U; 0 ≤ θ < 2π) , (21) or 1 z [(1 + Be iθ ) zD q f (z) − (1 + Ae iθ ) f (z)] ̸ = 0 (z ∈ U; 0 ≤ θ < 2π) . (22) Since f (z) ∗ z 1 − z = f (z) , f (z) ∗ z (1 − z) (1 − qz) = zD q f (z) . (23) Now from (23), we may write (22) as 1 z [f (z) ∗ ( (1 + Be iθ ) z (1 − z) (1 − qz) − (1 + Ae iθ ) z 1 − z )]


Introduction
Simply, ℎ-calculus or -calculus is ordinary classical calculus without the notion of limits.Here ℎ 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][4][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][7][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 A denote the class of functions of the form: which are analytic in the open unit disk U = { ∈ C : || < 1}.
Let S() and K() (0 ≤  < 1) denote the subclasses of A that consists, respectively, of starlike of order  and convex of order  in U (see [13]).If () and () are analytic in U, we say that () is subordinate to (), written () ≺ () if there exists a Schwarz function , which (by definition) is analytic in U with (0) = 0 and |()| < 1 for all  ∈ U, such that () = (()),  ∈ U. Furthermore, if the function  is univalent in U, then we have the following equivalence (see [14][15][16]): For functions  given by (1) and  given by the Hadamard product or convolution of  and  is defined by Let S[, ] and K[, ] denote the subclasses of the class A for −1 ≤  <  ≤ 1 which are defined by (see [17][18][19][20][21][22]) We note that For function  ∈ A given by (1) and 0 <  < 1, the derivative of a function  is defined by (see [1]) and   (0) =   (0).From ( 7), we deduce that where As  → 1, []  → .For a function ℎ() =   , we observe that where ℎ  is the ordinary derivative.Making use of the -derivative   (), we introduce the subclasses S  [, ] and K  [, ] of A for 0 <  < 1 and −1 ≤  <  ≤ 1 as follows: We note that lim From ( 11), we have In this paper, we investigate convolution properties, the necessary and sufficient condition and coefficient estimates for the classes S  [, ] and K  [, ] associated with the derivative   ().The motivation of this paper is to improve and generalize previously known results.
Proof.First suppose  defined by ( 1) is in the class S  [, ]; we have Since the function from the left-hand side of the subordination is analytic in U, it follows () ̸ = 0,  ∈ U * = U \{0}; that is, (1/)() ̸ = 0,  ∈ U, and this is equivalent to the fact that (18) holds for  = 1.From ( 19) according to the subordination of two analytic functions we say that there exists a function 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.
Taking  → 1 − 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].

Proof.
Set and we note that From the identity   () * () = () *   () (,  ∈ A) and the fact that the result follows from Theorem 1.
Taking  → 1 − in Theorem 1, we obtain the following result which improves the result of Aouf and Seoudy [23,Theorem 2].
Taking  → 1 − in Corollary 7, we obtain the following result which improves the convolution result of Silverman et al. [22,Theorem 2].

Theorem 9. A necessary and sufficient condition for the function 𝑓 defined by (1) to be in the class
Proof.From Theorem 1, we find that  ∈ S  [, ] if and only if for all  =   = ( − + )/( − ) and also for  = 1.The left-hand side of (38) can be written as Thus, the proof of The Theorem 9 is completed.
Taking  → 1 − in Theorem 9, we obtain the following result.

Corollary 10. A necessary and sufficient condition for the function 𝑓 defined by (1) to be in the class
Putting  = 1−2 (0 ≤  < 1) and  = −1 in Theorem 9, we obtain the following corollary.

Corollary 11. A necessary and sufficient condition for the function 𝑓 defined by (1) to be in the class
Taking  → 1 − in Corollary 11, we obtain the following corollary which improves the result of Ahuja [17, Corollary 1 when  = 0].

Corollary 12.
A necessary and sufficient condition for the function  defined by (1) to be in the class S() is that Abstract and Applied Analysis 5 Theorem 13.A necessary and sufficient condition for the function () defined by (1) to be in the class Proof.From Theorem 5, we find that  ∈ K  [, ] if and only if for all  =   = ( − + )/( − ) and also for  = 1.The left-hand side of (44) may be written as and this proves Theorem 13.
Taking  → 1 − in Theorem 13, we obtain the following result.

Corollary 15. A necessary and sufficient condition for the function 𝑓 defined by (1) to be in the class
Taking  → 1 − in Corollary 15, we obtain the following corollary which improves the result of Ahuja [17, Corollary 1 when  = 1].

Coefficient Estimates
As an application of Theorems 9 and 13, we next determine coefficient estimate and inclusion property for a function of