On the subordination under Bernardi operator

Abstract

Let $\mathcal{H}$ denote the class of analytic functions in the unit disc on the complex plane $\mathbf{C}$. Let $\mathcal{E}$ be a subclass of $\mathcal{H}$. If the operator $I:\mathcal{E}\rightarrow \mathcal{H}$ satisfies \begin{equation*} f(z)\prec g(z) \Rightarrow I[f](z)\prec I[g](z) \end{equation*} for all $f,g\in\mathcal{E}$, then it is called subordination-preserving operator on the class $\mathcal{E}$. In this work we consider the convexity of the Bernardi operator. We prove also that the Bernardi is the subordination-preserving operator on the class of starlike functions. The applications of main results are also presented.