Stable Functions of Janowski Type

TL;DR

This paper investigates the stability of certain Janowski-type functions under partial sums, establishing conditions under which these functions maintain stability relative to specific reference functions within the unit disk.

Contribution

The paper proves that a class of Janowski-type functions are stable with respect to particular reference functions, and identifies parameter ranges where stability does not hold.

Findings

Proves stability of $v_{\lambda}(A,B,z)$ with respect to $v_{\lambda}(0,B,z)$ for specified parameters.

Shows $v_{\lambda}(A,B,z)$ is not stable with respect to itself under certain conditions.

Identifies parameter ranges for stability and instability of these functions.

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 $f(z)=\displaystyle\sum_{k=0}^{\infty} a_kz^k$ is given by $s_n(f(z)) = \displaystyle\sum_{k=0}^{n} a_kz^k, \ n\in \mathbb{N} \cup \{0\}$. In this work, we consider the following function \begin{align*} v_{\lambda}(A,B,z)=\left(\frac{1+Az}{1+Bz}\right)^{\lambda} \end{align*} for $-1\leq B < A \leq 1$ and $0\leq \lambda \leq 1 $ for our investigation. The main purpose of this paper is to prove that $v_{\lambda}(A,B,z)$ is stable with respect to $\displaystyle v_{\lambda}(0,B,z)= \frac{1}{(1+Bz)^{\lambda}}$ for $0 < \lambda \leq 1 $ and $-1\leq B < A \leq 0$. Further, we prove that $v_{\lambda}(A,B,z)$ is not stable with respect to itself, when $0 < \lambda \leq 1 $ and $-1\leq B < A <0$. \end{abstract}