Regions of variability for generalized Janowski functions

Let $r\in \mathbb{C},$ $s\in [-1,0)$, $0\le \alpha <1.$ 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

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

where $\chi \left(0\right)=0$ and $ \left \vert \chi \left(\zeta \right)\right \vert < 1. $Inthisarticle,wefindtheregionsofvariability$ W_{\lambda }\left(\zeta _{0}, r, s, \alpha \right) $for$ \int \limits_{0}^{z_{0}}q\left(\rho \right) d\rho \ $when$ q $rangesovertheclass$ \mathcal{Q}_{\lambda }\left[r, s, \alpha \right] $ defined as

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

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.