Revisit Of Meromorphic Convex Functions
Vibhuti Arora, Vinayak M
TL;DR
The paper develops a Schwarzian-derivative based framework to study meromorphically convex functions of order $\alpha$ ($0\leq\alpha<1$). It introduces two key classes, $\mathcal{BC}(\alpha)$ for meromorphic convexity and $\mathcal{BC}_I(\alpha)$ for meromorphically inverse convexity, and derives a sharp sufficiency criterion: if $|S_f(z)| \le 2 q_\alpha(|z|)$ with $q_\alpha \in \mathcal{P}((1+\alpha)/2)$, then $f \in \mathcal{BC}(\alpha)$. The results also establish radius bounds for inverse convexity in the meromorphic setting, explore invariance and connections to starlike and inverse-convex classes ($\mathcal{BS}^*(\alpha)$ and $\mathcal{S}^*(\alpha)$), and present sharpness constructions via Koebe-type examples. Collectively, the work extends classical Schwarzian techniques to meromorphic contexts, provides new analytic-characterization tools for $\mathcal{BC}_I(\alpha)$, and clarifies the relationship between convex, starlike, and inverse-convex meromorphic mappings with explicit radius and constant-optimal results.
Abstract
Our primary aim is to explore a sufficient condition for the class of meromorphically convex functions of order $α$, where $0 \leq α< 1$. The investigation will focus on studying a class of continuous functions defined on $[0,1)$, and analyzing the properties of the Schwarzian norm of locally univalent meromorphic functions. Moreover, a new subclass of meromorphic functions is also introduced, and some of its characteristics are examined.