Duality on <i>q</i>-Starlike Functions Associated with Fractional <i>q</i>-Integral Operators and Applications

Abstract

In this paper, we make use of the Riemann–Liouville fractional q-integral operator to discuss the class Sq,δ*(α) of univalent functions for δ>0,α∈C−{0}, and 0|q|1. Then, we develop convolution results for the given class of univalent functions by utilizing a concept of the fractional q-difference operator. Moreover, we derive the normalized classes Pδ,qζ(β,γ) and Pδ,q(β) (0|q|1, δ≥0,0≤β≤1,ζ>0) of analytic functions on a unit disc and provide conditions for the parameters q,δ,ζ,β, and γ so that Pδ,qζ(β,γ)⊂Sq,δ*(α) and Pδ,q(β)⊂Sq,δ*(α) for α∈C−{0}. Finally, we also propose an application to symmetric q-analogues and Ruscheweh’s duality theory.

Keywords

• Riemann–Liouville
• <i>q</i>-analogue
• difference operator
• <i>q</i>-starlike functions
• duality principle
• dual set