Region of variablity for Bazilevic functions

Let $\mathcal{H}$ be the family of analytic functions defined in an open unit disk $\mathbb{U}\mathbb{=}\left\{z:|z|<1\right\}$ and

$$\mathcal{A}=\left\{f\in \mathcal{H}:f(0)=f^{{}^{\mathrm{\prime }}}(0)-1=0,(z\in \mathbb{U})\right\}.$$

For $A\in \mathbb{C},B\in [-1,0)$ and $\gamma \in \left(\frac{-\pi }{2},\frac{\pi }{2}\right),$ a function $h\in$ ${\mathcal{P}}_{\gamma }[\xi ,A;B]$ can be written as:

$$h(z)=\cos \gamma \frac{1+A\omega (z)}{1+B\omega (z)}+i\sin \gamma ,(\omega (0)=0,\left|\omega (z)\right|<1,z\in \mathbb{U}),$$

where $\xi ={\omega }^{\mathrm{\prime }}\left(0\right)\in \overline{\mathbb{U}}$. The family ${\mathcal{B}}_{\gamma }\left[\psi ,\xi ,\beta ,A;B\right]$ contains analytic functions $f$ in $\mathbb{U}$ such that

$$\frac{e^{i\gamma }z{f}^{\mathrm{\prime }}(z)}{{\left[f\left(z\right)\right]}^{1-\beta }{\left[\psi (z)\right]}^{\beta }}\in {\mathcal{P}}_{\gamma }[\xi ,A;B],$$

where $\psi$ is a starlike function. In this research, we find the region of variability denoted by ${\mathcal{V}}_{\gamma }[\psi ,z_0,\xi ,A;B]$ for $f\left(z_0\right)$, where $f$ is ranging over the family ${\mathcal{B}}_{\gamma }\left[\psi ,\xi ,\beta ,A;B\right]$ for any fixed $z_0\in \mathbb{U}$ and $\xi \in \overline{\mathbb{U}}.$