q-analogue of differential subordinations

Abstract

In this article, we study differential subordnations in q-analogue. Some properties of analytic functions in q-analogue associated with cardioid domain and limacon domain are considered. In particular, we determine conditions on a such that 1 + α z∂qh(z)/(h(z))n (n = 0, 1, 2, 3) are subordinated by Janowski functions and h (z) ≺ 1 + 4/3 z + 2/3 z2. We also consider the same implications such that h (z) ≺ 1 + √2z + 1/2 z2. We apply these results on analytic functions to find sufficient conditions for q-starlikeness related with cardioid and limacon.