Abstract
In this paper, the authors introduced certain subclasses β-uniformly q-starlike and β-uniformly q-convex functions of order α involving the q-derivative operator defined in the open unit disc. Coefficient bounds were also investigated.
Highlights
The q-analysis is a generalization of the ordinary analysis
The authors introduced certain subclasses β-uniformly q-starlike and β-uniformly q-convex functions of order α involving the q-derivative operator defined in the open unit disc
New subclasses of analytic functions associated with q-derivative operators are introduced and discussed, see for example [4, 6,7,8,9,10,11,12,13,14,15,16,17,18]
Summary
The q-analysis is a generalization of the ordinary analysis. The application of the q-calculus was first introduced by Jackson [1,2,3]. Motivated by the importance of q-analysis, in this paper, we introduce the classes of β -uniformly q-starlike and β-uniformly q-convex functions defined by the q-derivative operator in the open unit disc, as a generalization of β-uniformly starlike and β-uniformly convex functions. Since we note that 1⁄2nq ⟶ n as q ⟶ 1−, in view of equation (4), DqhðzÞ ⟶ h′ðzÞ as q ⟶ 1−, where h′ðzÞ denotes the ordinary derivative of the function hðzÞ with respect to z. The classes of β-uniformly starlike functions of order α and β-uniformly convex functions of order α, denoted by SDðα, βÞ and KDðα, βÞ, respectively, are defined as follows [20]:. We introduce the classes of β-uniformly q-starlike and β-uniformly q-convex functions of order α, denoted by Sqðα, βÞ and UCV qðα, βÞ, respectively. We obtain the coefficient bounds of the functions belonging to these classes
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