Regions of variability for generalized Janowski functions

Let $ r\in \mathbb{C}, $ $ s\in \lbrack -1, 0) $, $ 0\leq \alpha < 1. $ Then, $ \mathcal{Q}\left[r, s, \alpha \right] $ stands for the set of analytic functions $ q $ that is within the open unit disk $ E $, with $ q\left(0\right) = 1, $ and satisfies the explicit representation

$ \begin{equation*} q\left( \zeta \right) = \frac{1+\left( \left( 1-\alpha \right) r+\alpha s\right) \chi \left( \zeta \right) }{1+s\chi \left( \zeta \right) }, \end{equation*} $

where $ \chi \left(0\right) = 0 $ and $ \left \vert \chi \left(\zeta \right)\right \vert < 1. $ In this article, we find the regions of variability $ W_{\lambda }\left(\zeta _{0}, r, s, \alpha \right) $ for $ \int \limits_{0}^{z_{0}}q\left(\rho \right) d\rho \ $when $ q $ ranges over the class $ \mathcal{Q}_{\lambda }\left[r, s, \alpha \right] $ defined as

$ \begin{equation*} \mathcal{Q}_{\lambda }\left[ r, s, \alpha \right] = \left \{ q\in \mathcal{Q} \left[ r, s, \alpha \right] :q^{\prime }\left( 0\right) = \left( \left( 1-\alpha \right) \left( r-s\right) \right) \lambda \right \} \end{equation*} $

for any fixed $ \zeta _{0}\in E $ and $ \lambda \in \overline{E} $. As a corollary, the region of variability appears for the alternate sets of parameters as well.