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.
Highlights
Introduction andDefinitions a function f is defined byLet Ap denote the family of all meromorphic p-valent functions f that are analytic in the punctured disc D = fz ∈ C: 0 < jzj < 1g and obeying the normalization f ðz Þ = ∞+ 〠 an z n, p z n=p+1 z ∈ D: ð1Þ f ðz Þ ∈ MC p ðαÞ ⇔ Re f ′ ðz Þ < −α: ð2ÞFor 0 < q < 1, the q-difference operator or q-derivative of f ðqz Þ − f ðz Þ, z ðq − 1 Þ z ≠ 0, q ≠ 1: ð3ÞIt can be seen that for n ∈ N, where N stands for the set of natural numbers and z ∈ D
It can be seen that for n ∈ N, where N stands for the set of natural numbers and z ∈ D, (
Let MC p ðαÞ denote the well-known family of meromorphic p-valent convex functions of order αð0 ≤ α < pÞ
Summary
Let MC p ðαÞ denote the well-known family of meromorphic p-valent convex functions of order αð0 ≤ α < pÞ and defined as. In somewhat similar way, Mohammed and Darus [12] introduced a generalized operator along with investigating a class of functions relating to q. Arif and Ahmad defined a new q-differential operator for meromorphic multivalent functions and investigated classes related to q-meromorphic starlike and convex functions in their articles [14, 15]. We introduce a new q-differential operator for meromorphic functions and use this operator to define and study some properties of a new family of meromorphic multivalent functions associated with circular domain. ÐA − BÞ1⁄2p, q2 ðA − BÞ1⁄2p, q2 /qp − ∑n=p+1 ð1 + 1⁄2p, qμ + μqp 1⁄2n, qÞm A1⁄2p, q1⁄2n, q + Bqp 1⁄2n, q2 jan j. Combining (50) and (51), we readily get the coefficient estimates (30)
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