Two normality criteria and counterexamples to the converse of the bloch's principle

TL;DR

This paper introduces two new criteria for the normality of meromorphic function families and provides counterexamples to the converse of Bloch's principle, advancing understanding in complex analysis.

Contribution

It extends existing normality criteria to broader classes of differential polynomials and offers counterexamples to a key conjecture in the field.

Findings

Extended normality criterion to larger classes of differential polynomials

Provided counterexamples to the converse of Bloch's principle

Enhanced understanding of normal families in complex analysis

Abstract

In this paper, we prove two normality criteria for a family of meromorphic functions. The first criterion extends a result of Fang and Zalcman[Normal families and shared values of meromorphic functions II, Comput. Methods Funct. Theory, 1(2001), 289 - 299] to a bigger class of differential polynomials whereas the second one leads to some counterexamples to the converse of the Bloch's principle.