Abstract
Making use of fractional q-calculus operators, we introduce a new subclass ℳq(λ,γ,k) of starlike functions and determine the coefficient estimate, extreme points, closure theorem, and distortion bounds for functions in ℳq(λ,γ,k). Furthermore we discuss neighborhood results, subordination theorem, partial sums, and integral means inequalities for functions in ℳq(λ,γ,k).
Highlights
Introduction and PreliminariesWe recall the definitions of fractional q-calculus operators of complex valued function f(z)
Introduction and PreliminariesDenote by A the class of functions of the form ∞f (z) = z + ∑anzn (1)n=2 which are analytic and univalent in the open disc U = {z : |z| < 1} and normalized by f(0) = 0 = f(0) − 1
The fractional q-calculus operator is the extension of the ordinary fractional calculus in the qtheory
Summary
We recall the definitions of fractional q-calculus operators of complex valued function f(z). Further the q-derivative and q-integral of functions f defined on the subset of C are, respectively, given by f (z) − f (zq). The fractional q-derivative operator of order μ is defined for a function f(z) by. Motivated by the earlier works of Goodman [5] and Rønning [6, 7] in this paper we define the following new subclass of k-starlike functions of order γ based on the q-fractional operator. For 0 ≤ λ < 1, 0 ≤ γ < 1, μ < 2, and k ≥ 0, we let Mq(λ, γ, k) be the subclass of T consisting of functions of the form (2) and satisfying the analytic criterion.
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