AIMS Mathematics, 2020, 5(4): 3365-3377. doi: 10.3934/math.2020217

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Regions of variability for a subclass of analytic functions

1 Department of Mathematics, Government College University, Faisalabad, Pakistan
2 Department of Mathematics, Abbottabad University of Science and Technology Abbottabad, Pakistan
3 Department of Mathematics, Yangzhou University, Yangzhou 225002, China

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 f(0)=0=f'(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 $\left\vert w\left(z\right) \right\vert <1.$ 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.
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