Some results on generalized Cesàro stable of Janowski function

Let \(g_1\) $g_1$ and \(g_2 \) $g_2$ be any two analytic functions defined in the unit disc which are normalized by the condition \(g_1(0)=1=g_2(0)\) $g_1(0)=1=g_2(0)$ and \(\sigma _n^{b-1,c}(z)\) ${\sigma }_n^{b-1,c}(z)$ be the n th Cesàro mean of type \((b-1,c)\) $(b-1,c)$ for \(1+b>c>0\) $1+b>c>0$ . Then \(g_1\) $g_1$ is generalized Cesàro stable with respect to \(g_2\) $g_2$ , whenever \(\begin{aligned} \dfrac{\sigma _n^{b-1,c}(g_1,z)}{g_1(z)}\prec \dfrac{1}{g_2(z)}\quad \ (z \in \Delta ,\ n \in \mathbb {N}_0), \end{aligned}\) $$\begin{array}{c}\hfill {\displaystyle \frac{{\sigma }_n^{b-1,c}(g_1,z)}{g_1(z)}}\prec {\displaystyle \frac{1}{g_2(z)}}(z\in \mathrm{\Delta },n\in {\mathbb{N}}_0),\end{array}$$ where \(\sigma _n^{b-1,c}(g,z) = \dfrac{1}{B_n} \sum _{j=0}^{n} B_{n-j} b_j z^j = \sigma _n^{b-1,c}(z) * g(z)\) ${\sigma }_n^{b-1,c}(g,z)={\displaystyle \frac{1}{B_n}}{\sum }_{j=0}^n{B}_{n-j}{b}_j{z}^j={\sigma }_n^{b-1,c}(z)\ast g(z)$ . The main aim of this article is to prove that \(\left( {(Az+1)}/{(Bz+1)}\right) ^\eta \) ${\left((Az+1),/,(Bz+1)\right)}^{\eta }$ is generalized Cesàro stable with respect to \((1/(Bz+1)^{\eta })\) $(1/{(Bz+1)}^{\eta })$ but not with respect to itself for \(-1 \le B < A \le 0\) $-1\le B<A\le 0$ and \(0<\eta \le 1\) $0<\eta \le 1$ . As an application, we obtain new and existing results on Cesàro stability and stability.