On Stable Functions

Abstract

Let $\mathcal {A}_{0}$ denote the class of analytic functions f in the unit disk ${\mathbb {D}} = \{ z \in {\mathbb {C}} : |z| < 1\}$ normalized by $f(0)=1$. For $f(z) = \sum \nolimits _{k=0}^{\infty } a_{k} z^{k}$, the nth partial sum $s_{n} (f,z)$ of f is defined by $s_{n} (f,z) = \sum \nolimits _{k=0}^{n} a_{k} z^{k}$, $n=0,1,\ldots$. A function $f \in \mathcal {A}_{0}$ is said to be stable with respect to $g \in \mathcal {A}_{0}$ if

$$\begin{aligned} \frac{s_{n} (f,z)}{f(z)} \prec \frac{1}{g(z)}, \quad z \in {\mathbb {D}} \end{aligned}$$

holds for all $n \in {\mathbb {N}}$. In the present paper, we consider the following function

$$\begin{aligned} v_{\lambda } (\alpha , z) := \left( \frac{1+(1-2\alpha ) z}{1-z}\right) ^{\lambda }, \end{aligned}$$

for $0\le \alpha < 1$ and $0\le \lambda \le 1$. The aim of this paper is to prove that $v_{\lambda }(\alpha ,z)$ is stable with respect to $f_{\lambda } (z):= 1/(1-z)^{\lambda }$ for $0<\lambda \le 1$ and $1/2 \le \alpha < 1$. Also, we prove that $v_{\lambda }(\alpha ,z)$ is not stable with respect to itself when $1/2< \alpha <1$ and $0 <\lambda \le 1$. Finally, we end this paper with two conjectures.