On Geometrical Properties of Certain Analytic functions

Abstract

We introduce the class of analytic functions $$\mathcal{F}(\psi ):=\left\{f\in \mathcal{A}:\left(\frac{z{f}^{\prime }(z)}{f(z)}-1\right)\prec \psi (z),\psi (0)=0\right\},$$ where $\psi$ is univalent and establish the growth theorem with some geometric conditions on $\psi$ and obtain the Koebe domain with some related sharp inequalities. Note that functions in this class may not be univalent. As an application, we obtain the growth theorem for the complete range of $\alpha$ and $\beta$ for the functions in the classes $\mathcal{BS}(\alpha ):=\{f\in \mathcal{A}:(z{f}^{\prime }(z)/f(z))-1\prec z/(1-\alpha z^2),\alpha \in [0,1)\}$ and ${\mathcal{S}}_{cs}(\beta ):=\{f\in \mathcal{A}:(z{f}^{\prime }(z)/f(z))-1\prec z/((1-z)(1+\beta z)),\beta \in [0,1)\}$, respectively which improves the earlier known bounds. The sharp Bohr-radii for the classes $S(\mathcal{BS}(\alpha ))$ and $\mathcal{BS}(\alpha )$ are also obtained. A few examples as well as certain newly defined classes on the basis of geometry are also discussed.