On a Class of Certain Non-univalent Functions

Abstract

In this paper, we introduce a family of analytic functions given by \(\psi _{A,B}(z):= \dfrac{1}{A-B}\log {\dfrac{1+Az}{1+Bz}},\) which maps univalently the unit disk onto either elliptical or strip domains, where either \(A=-B=\alpha\) or \(A=\alpha e^{i\gamma }\) and \(B=\alpha e^{-i\gamma }\) (\(\alpha \in (0,1]\) and \(\gamma \in (0,\pi /2]\)). We study a class of non-univalent analytic functions defined by \({{\mathcal {F}}}[A,B]:=\left\{ f\in {{\mathcal {A}}}:\left( \dfrac{zf'(z)}{f(z)}-1\right) \prec \psi _{A,B}(z)\right\}\). Further, we investigate various characteristic properties of \(\psi _{A,B}(z)\) as well as functions in the class \({{\mathcal {F}}}[A,B]\) and obtain the sharp radius of starlikeness of order \(\delta\) and univalence for the functions in \({{\mathcal {F}}}[A,B]\). Also, we find the sharp radii for functions in \({{{\mathcal {B}}}}{{{\mathcal {S}}}}(\alpha ):=\{f\in {{\mathcal {A}}}:zf'(z)/f(z)-1\prec z/(1-\alpha z^2),\;\alpha \in (0,1)\}\), \({{\mathcal {S}}}_{cs}(\alpha ):=\{f\in {{\mathcal {A}}}:zf'(z)/f(z)-1\prec z/((1-z)(1+\alpha z)),\;\alpha \in (0,1)\}\), and others to be in the class \({{\mathcal {F}}}[A,B].\)