Regions of variability for a subclass of analytic functions

Let $A\in \mathbb{C}, $ $B\in \lbrack -1, 0)\ $and $\alpha \in \left(-\frac{\pi }{2}, \frac{\pi }{2% }\right) $. Then $C_{\alpha }\left[ A, B\right] $ denotes the class of analytic functions $f$ in the open unit disc with \lt i \gt f \lt /i \gt (0) = 0 = \lt i \gt f \lt /i \gt '(0)-1 such that $ \begin{equation*} e^{i\alpha }\left(1+\frac{zf^{\prime \prime }\left(z\right) }{f^{\prime }\left(z\right) }\right) = \cos \alpha p\left(z\right) +i\sin \alpha, \end{equation*} $ with $ \begin{equation*} p\left(z\right) = \frac{1+Aw\left(z\right) }{1+Bw\left(z\right) }, \end{equation*} $ where $w\left(0\right) = 0$ and $\left\vert w\left(z\right) \right\vert \lt 1.$ Region of variability problems provides accurate information about a class of univalent functions than classical growth distortion and rotation theorems. In this article we find the regions of variability $V_{\lambda }\left(z_{0}, A, B\right) $ for $\log f^{\prime }\left(z_{0}\right) \ $when $f$ ranges over the class $C_{\alpha }\left[ \lambda, A, B\right] $ defined as $ \begin{equation*}{{C}_{\alpha }}\left[ \lambda, A, B \right] = \left\{ f\in {{C}_{\alpha }}\left[ A, B \right]:f''\left(0 \right) = \left(A-B \right){{e}^{-i\alpha }}\cos \alpha \right\}\end{equation*} $ for any fixed $z_{0}\in E$ and $\lambda \in \overline{E}$. As a consequence, the regions of variability are also illustrated graphically for different sets of parameters.