On Geometrical Properties of Certain Analytic functions

Abstract

Let $\psi$ be univalent, we introduce a newly defined class of analytic functions, given by

$$\begin{aligned} {\mathcal {F}}(\psi ):= \left\{ f\in {\mathcal {A}}: \left( \frac{zf'(z)}{f(z)}-1\right) \prec \psi (z),\ \psi (0)=0 \right\} . \end{aligned}$$

Note that functions in this class may not be univalent. Further, we establish the growth theorem with some geometrical conditions on $\psi$ and obtain the Koebe domain with some related sharp inequalities. As an application, for the complete range of $\alpha$ and $\beta$, we obtain the growth theorem for functions in the classes ${{\mathcal {B}}}{{\mathcal {S}}}(\alpha ):= \{f\in {\mathcal {A}} : ({zf'(z)}/{f(z)})-1 \prec {z}/{(1-\alpha z^2)},\ \alpha \in [0,1) \}$ and ${\mathcal {S}}_{\hbox {cs}}(\beta ):= \{f\in {\mathcal {A}} : ({zf'(z)}/{f(z)})-1 \prec {z}/({(1-z)(1+\beta z)}),\ \beta \in [0,1) \}$, respectively, which in fact improves the earlier known bounds. The sharp Bohr radii for the classes $S({{\mathcal {B}}}{{\mathcal {S}}}(\alpha ))= \{g=\sum _{k=1}^{\infty }b_k z^k : g\prec f,\ f\in {{\mathcal {B}}}{{\mathcal {S}}}(\alpha ) \}$ and ${{\mathcal {B}}}{{\mathcal {S}}}(\alpha )$ are also obtained. A few examples as well as certain newly defined classes on the basis of geometry are also discussed.