Radius of exponential convexity of certain subclass of analytic functions

Varma, S. Sunil, Rosy, Thomas and Vadivelan, U.

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creative_2020_29_1_109_112

DOI: https://doi.org/10.37193/CMI.2020.01.13

Issue no: Vol 29/2020 no. 1 Tags: radius of exponential convexity, univalent functions

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Let $\mathcal{S}$ be the class of analytic normalized univalent functions defined on the unit disk\linebreak $\Delta = \lbrace z \in \mathbb{C} : |z|<1 \rbrace$ and $f$ , $g$ be two functions in $\mathcal{S}$ satisfying $\displaystyle Re \bigg ( \frac{zf'(z)e^{\alpha f(z)}}{g(z)} \bigg ) >0$ and $\displaystyle\bigg | \frac{zf'(z)e^{\alpha f(z)}}{g(z)} - 1 \bigg | <1$ , $\alpha \in \mathbb{C} \setminus \lbrace 0 \rbrace$ . We determine the radius of exponential convexity of $f \in \mathcal{S}$ whenever $g$ satisfies $(i) \ Re \dfrac{g(z)}{z} >0$ \ $(ii) \ Re \dfrac{g(z)}{z} > 1/2$ .