Gehring-Hayman Inequality for Meromorphic Univalent Mappings

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 ${\text{IT}}^{-}$ of the unit circle. In this article, we prove that the ratio of the length of the image of the vertical diameter $\text{IA}$ of the unit disk to the length of the image of ${\text{IT}}^{-}$ under the mapping $f$ is bounded by a constant depending only on $p\mathrm{.}$ Next, we extend this result by considering any hyperbolic geodesic and any Jordan curve in $\text{D}$ 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. $\mathbf{\text{199}}$ (2025), # 103583].