On q-ANALOGUE of Differential Subordination Associated with Lemniscate of Bernoulli

Abstract

This article comprises the study of differential subordination with analogue of $q$ -derivative. It includes the sufficient condition on $\gamma$ for $1+\left(\gamma \partial z_{q}h\left(z\right)/h^n\left(z\right)\right)$ to be subordinated by $\left(1+Az/1+Bz\right)$ , $-1\le B<A\le 1$ , and implies that $h\left(z\right)\prec \sqrt{1+z}$ , where $h\left(z\right)$ is the analytic function in the open unit disk. Moreover, certain sufficient conditions for $q$ -starlikeness of analytic functions related with lemniscate of Bernoulli are determined.