Gehring-Hayman Inequality for Meromorphic Univalent Mappings

TL;DR

This paper extends the classical Gehring-Hayman inequality to meromorphic univalent functions, establishing bounds on image lengths of certain curves and confirming a recent conjecture in the field.

Contribution

It proves a new inequality for meromorphic univalent functions, extending classical results and confirming a conjecture by Bhowmik and Maity.

Findings

Bound on the ratio of image lengths depending only on pole position

Extension of the inequality to hyperbolic geodesics and Jordan curves

Confirmation of the conjecture by Bhowmik and Maity

Abstract

Let $f$ be a meromorphic univalent function on the open unit disk having a simple pole at $p\in (0,1)$ that extends continuously to the left half $\IT^{-}$ of the unit circle. In this article, we prove that the ratio of the length of the image of the vertical diameter $\IA$ of the unit disk to the length of the image of $\IT^{-}$ under the mapping $f$ is bounded by a constant depending only on $p.$ Next, we extend this result by considering any hyperbolic geodesic and any Jordan curve in $\D$ sharing the same endpoints. These results extend the classical Gehring-Hayman inequality to meromorphic univalent functions and also prove a conjecture posed by Bhowmik and Maity [Bull. Sci. Math. \textbf{199} (2025), \# 103583].