Duality on q-Starlike Functions Associated with Fractional q-Integral Operators and Applications.

In this paper, we make use of the Riemann-Liouville fractional q-integral operator to discuss the class S-q,S-delta* (a) of univalent functions for delta > 0, alpha is an element of C - {0}, and 0 < vertical bar q vertical bar < 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 P-delta,q(zeta) (beta, gamma) and P-delta,(q) (beta) (0 < vertical bar q vertical bar < 1, delta >= 0, 0 <= beta <= 1, zeta > 0) of analytic functions on a unit disc and provide conditions for the parameters q, delta, zeta, beta, and gamma so that P-delta,q(zeta) (beta, gamma) subset of S-q,S-delta* (alpha) and P-delta,P-q(beta) subset of S-q,S-delta* (alpha) for a is an element of C - {0}. Finally, we also propose an application to symmetric q-analogues and Ruscheweh's duality theory.