Radii of convexity associated with various subclasses of analytic functions for functions related to Kaplan classes

Abstract

A normalized analytic function $f$ defined on the open unit disc $\mathbb{D}$ is called Ma-Minda convex if $ 1+zf''(z)/f'(z)$ is subordinate to the function $\varphi$. For $ 0\leqslant\alpha\leqslant\beta$, the Kaplan class $\mathcal{K}(\alpha,\beta)$ of type $\alpha$ and $\beta$ consists of normalized analytic functions of the form $p^\alpha g$ defined on $\mathbb{D}$ where $p$ with $p(0)=1$ is an analytic function taking values in the right half-plane and $g$ is an analytic function with $g(0)=1$ satisfying Re$(zg'(z)/g(z))>(\alpha-\beta)/2$. For functions $f$ with $f'\in \mathcal{K}(\alpha,\beta)$, we obtain the radius of Ma-Minda convexity for various choices of $\varphi$. The radius of lemniscate convexity, lune convexity, nephroid convexity, exponential convexity and several other radius estimates are examined. The results obtained are sharp.

Keywords

• univalent functions
• starlike functions
• convex functions
• close–to–convex functions
• subordination
• Kaplan class
• radius of convexity

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