On q-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.

A Study on Certain Subclasses of Analytic Functions Involving the Jackson q-Difference Operator

“…We observe that when q → 1 − , we have R n q f (z) = D n f (z). For more details on the qanalogue Ruschewewh differential operators, see [24][25][26][27]. Now, we define the function ϕ(a, q, z) by…”