Abstract
In the present paper, by using the concept of convolution and q-calculus, we define a certain q-derivative (or q-difference) operator for analytic and multivalent (or p-valent) functions. This presumably new q-derivative operator is an extension of the known q-analogue of the Ruscheweyh derivative operator. We also give some interesting applications of this q-derivative operator for multivalent functions by using the method of differential subordination. Relevant connections with a number of earlier works on this subject are also pointed out.
Highlights
Definition 2 For two analytic functions fj (j = 1, 2) in U, the function f1 is said to be subordinate to the function f2, which is written as follows: f1 âș f2 or f1(z) âș f2(z) (z â U), if we can find a Schwartz function w, analytic in U, with w(0) = 0 and w(z) < 1, such that f1(z) = f2 w(z)
Aldweby [2] and SokĂłl [21] studied some classes of analytic functions defined by means of the q-analogue of Ruscheweyhâs derivative operator
In order to define this extended q-analogue of Ruscheweyhâs derivative operator, we use the concepts of the Hadamard product
Summary
Definition 1 The Hadamard product or convolution of the following two functions fj(z) â A(p) (j = 1, 2). Definition 2 For two analytic functions fj (j = 1, 2) in U, the function f1 is said to be subordinate to the function f2, which is written as follows: f1 âș f2 or f1(z) âș f2(z) (z â U), if we can find a Schwartz function w, analytic in U, with w(0) = 0 and w(z) < 1, such that f1(z) = f2 w(z). If the function f2 is univalent in U, the following equivalence relation holds true: f1(z) âș f2(z) (z â U) ââ f1(0) = f2(0) and f1(U) â f2(U). Which are analytic in U and satisfy the following inequalities: Ï(z) > ÎČ (0 ÎČ < 1)
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