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 α such that 1 + α z ∂ q h z h z n ( n = 0 , 1 , 2 , 3 ) are subordinated by Janowski functions and h z ≺ 1 + 4 3 z + 2 3 z 2 . We also consider the same implications such that h z ≺ 1 + 2 z + 1 2 z 2 . We apply these results on analytic functions to find sufficient conditions for q-starlikeness related with cardioid and limacon.

Keywords

• <i>q</i>-calculus
• differential subordination
• Janowski function
• cardioid domain
• limacon domain