Some Normality Criteria and a Counterexample to the Converse of Bloch's Principle

TL;DR

This paper develops new normality criteria for families of meromorphic functions using value distribution and differential polynomials, and provides a counterexample to the converse of Bloch's principle.

Contribution

It introduces novel normality criteria involving differential polynomials and shares a counterexample to the converse of Bloch's principle.

Findings

Established new normality criteria for meromorphic functions.

Extended and improved existing results in value distribution theory.

Provided a counterexample to the converse of Bloch's principle.

Abstract

In this paper we continue our earlier investigations on normal families of meromorphic functions\cite{CS2}. Here, we prove some value distribution results which lead to some normality criteria for a family of meromorphic functions involving the sharing of a holomorphic function by more general differential polynomials generated by members of the family and get some recently known results extended and improved. In particular, the main result of this paper leads to a counterexample to the converse of Bloch's principle.