On <i>q</i>-Uniformly Mocanu Functions

Abstract

Let f be analytic in open unit disc E = { z : | z | < 1 } with f ( 0 ) = 0 and f ′ ( 0 ) = 1 . The q-derivative of f is defined by: D q f ( z ) = f ( z ) − f ( q z ) ( 1 − q ) z , q ∈ ( 0 , 1 ) , z ∈ B − { 0 } , where B is a q-geometric subset of C . Using operator D q , q-analogue class k − U M q ( α , β ) , k-uniformly Mocanu functions are defined as: For k = 0 and q → 1 − , k − reduces to M ( α ) of Mocanu functions. Subordination is used to investigate many important properties of these functions. Several interesting results are derived as special cases.

Keywords

• <i>q</i>-calculus
• <i>q</i>-starlike
• uniformly convex
• subordination
• Mocanu functions
• <i>q</i>-Ruscheweyh derivative