Applications of a New $q$-Difference Operator in Janowski-Type Meromorphic Convex Functions

Abstract

The main aim of the present article is the introduction of a new differential operator in q -analogue for meromorphic multivalent functions which are analytic in punctured open unit disc. A subclass of meromorphic multivalent convex functions is defined using this new differential operator in q -analogue. Furthermore, we discuss a number of useful geometric properties for the functions belonging to this class such as sufficiency criteria, coefficient estimates, distortion theorem, growth theorem, radius of starlikeness, and radius of convexity. Also, algebraic property of closure is discussed of functions belonging to this class. Integral representation problem is also proved for these functions.

Univalent Functions by Means of Chebyshev Polynomials

“…For analytic functions f(z) and F(z) in D, we say f is subordinate to F, written f≺F, if there exists a function w analytic in D, with w(0) � 0 and |w(z)| < 1 such that f(z) � F(w(z)), see [3,12]. If F is univalent, then…”

“…where t � cos θ and n is the degree of polynomial. For more details, one may refer to [1][2][3][4][5][6]. e polynomials in (4) are connected by the following relations:…”

Starlike Functions of the Miller–Ross-Type Poisson Distribution in the Janowski Domain

“…It has been noted that this formalism facilitates more mathematical investigation and aids in a better understanding of the symmetric and geometric characteristics of such operators. The work of [13,14] makes it easy to see the significance of convolution in the theory of operators. We consider the linear operator…”

Properties of q-Starlike Functions Associated with the q-Cosine Function

“…Recently, Ahmad et al [8], introduced a class of meromorphic Janowskitype multivalent q-starlike functions involving the q-differential operator and discussed some of its important geometric properties. Moreover, Ahmad et al [9] introduced a new q-differential operator and described some applications to the class of convex functions. Furthermore, the q-trigonometric functions are given in [10,11].…”