Stable Functions of Janowski Type

Abstract

A function $f\in {\mathcal{A}}_1$ is said to be stable with respect to $g\in {\mathcal{A}}_1$ if \begin{align*} \frac{s_n(f(z))}{f(z)} \prec \frac{1}{g(z)}, \qquad z\in\mathbb{D}, \end{align*} holds for all $n\in \mathbb{N}$ where ${\mathcal{A}}_1$ denote the class of analytic functions $f$ in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ normalized by $f(0)=1$. Here $s_n(f(z))$, the $n^{th}$ partial sum of $f(z)=\sum _{k=0}^{\infty }{a}_k{z}^k$ is given by $s_n(f(z))=\sum _{k=0}^n{a}_k{z}^k,n\in \mathbb{N}\cup \{0\}$. In this work, we consider the following function \begin{align*} v_λ(A,B,z)=\left(\frac{1+Az}{1+Bz}\right)^λ \end{align*} for $-1\le B<A\le 1$ and $0\le \lambda \le 1$ for our investigation. The main purpose of this paper is to prove that $v_{\lambda }(A,B,z)$ is stable with respect to $v_{\lambda }(0,B,z)=\frac{1}{(1+Bz{)}^{\lambda }}$ for $0<\lambda \le 1$ and $-1\le B<A\le 0$. Further, we prove that $v_{\lambda }(A,B,z)$ is not stable with respect to itself, when $0<\lambda \le 1$ and $-1\le B<A<0$. \end{abstract}

9 pages, 1 figure