Region of variablity for Bazilevic functions

Abstract

Let $ \mathcal{H} $ be the family of analytic functions defined in an open unit disk $ \mathbb{U = }\left \{ z:|z| < 1\right \} $ and $ \mathcal{A} = \left \{ f\in \mathcal{H}:f(0) = f^{^{\prime}}(0)-1 = 0, { \ \ \ \ \ }(z\in \mathbb{U})\right \} . $ For $ A\in \mathbb{C}, B\in \lbrack-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, \qquad (\omega(0) = 0, \left \vert \omega(z)\right \vert <1, z\in \mathbb{U}), $ where $ \xi = \omega^{\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}zf^{\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}}. $

Keywords

• spirallike and bazilevičfunctions
• schwarz lemma