Bloch and Landau constants for meromorphic functions

Abstract

Let $\mathcal{M}_1(\lambda)$ be the class of all meromorphic functions $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}\}: |z|<1$ having a simple pole at $\lambda \in \overline{\mathbb{D}} \setminus \{0\}$ and satisfying the normalization $f'(0)=1$. Let $B(\lambda)$ and $L(\lambda)$ denote the Bloch and Landau constants, respectively, for this class. In this article, we first show that the Bloch constant $B(1)$ and the Landau constant $L(1)$ are infinite. Using these results and a conformal mapping technique, we establish that $B(p)$ and $L(p)$ are likewise infinite for any $p \in (0,1)$, thereby refuting a recent conjecture. Finally, we extend our study to the class of meromorphic functions having two simple poles and prove that their associated Bloch and Landau constants also remain infinite.