Bloch and Landau constants for meromorphic functions

TL;DR

This paper investigates the Bloch and Landau constants for classes of meromorphic functions with poles in the unit disk, proving these constants are infinite for certain classes and refuting a recent conjecture.

Contribution

It demonstrates that the Bloch and Landau constants are infinite for specific classes of meromorphic functions, including those with one or two poles, challenging previous conjectures.

Findings

Bloch and Landau constants are infinite for functions with a pole at 1.

Constants are also infinite for functions with poles inside the disk.

Refutes a recent conjecture about the finiteness of these constants.

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.