Some results on generalized Cesàro stable of Janowski function

Abstract

Let \(g_1\) and \(g_2 \) be any two analytic functions defined in the unit disc which are normalized by the condition \(g_1(0)=1=g_2(0)\) and \(\sigma _n^{b-1,c}(z)\) be the n th Cesàro mean of type \((b-1,c)\) for \(1+b>c>0\). Then \(g_1\) is generalized Cesàro stable with respect to \(g_2\), whenever

$$\begin{aligned} \dfrac{\sigma _n^{b-1,c}(g_1,z)}{g_1(z)}\prec \dfrac{1}{g_2(z)}\quad \ (z \in \Delta ,\ n \in \mathbb {N}_0), \end{aligned}$$

where \(\sigma _n^{b-1,c}(g,z) = \dfrac{1}{B_n} \sum _{j=0}^{n} B_{n-j} b_j z^j = \sigma _n^{b-1,c}(z) * g(z)\). The main aim of this article is to prove that \(\left( {(Az+1)}/{(Bz+1)}\right) ^\eta \) is generalized Cesàro stable with respect to \((1/(Bz+1)^{\eta })\) but not with respect to itself for \(-1 \le B < A \le 0\) and \(0<\eta \le 1\). As an application, we obtain new and existing results on Cesàro stability and stability.