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Normality through partial sharing of sets with differential polynomials

Kuldeep Singh Charak, Nikhil Bharti, Anil Singh

TL;DR

The paper develops normality criteria for families of meromorphic functions by using partial sharing of sets with differential polynomials, extending prior results that linked normality to sharing with derivatives. It introduces a differential-polynomial framework $P[f]=\sum_{i=1}^{m}a_i M_i[f]$, with normalized monomials $M_i$ and a key multiplicity parameter $\alpha$, and proves that if $f-a$ has zeros of multiplicity at least $\alpha+1$ for all $a\in S_1$ and $P[f](z)\in S_1$ implies $f(z)\in S_2$, then the family is normal in $D$; analogous criteria are established for normal meromorphic functions. The results generalize several known theorems (e.g., in the derivative setting) and answer questions about replacing derivatives with differential polynomials and about partial sharing, using a toolkit that combines Zalcman–Pang rescaling with Nevanlinna theory. Collectively, these findings broaden the scope of normality criteria in complex analysis and supply a unified approach to partial sharing with differential polynomials.

Abstract

This article aims at finding sufficient conditions for a family of meromorphic functions to be normal by involving partial sharing of sets with differential polynomials. Moreover, corresponding results for normal meromorphic functions are also established which improve and generalize many known results.

Normality through partial sharing of sets with differential polynomials

TL;DR

The paper develops normality criteria for families of meromorphic functions by using partial sharing of sets with differential polynomials, extending prior results that linked normality to sharing with derivatives. It introduces a differential-polynomial framework , with normalized monomials and a key multiplicity parameter , and proves that if has zeros of multiplicity at least for all and implies , then the family is normal in ; analogous criteria are established for normal meromorphic functions. The results generalize several known theorems (e.g., in the derivative setting) and answer questions about replacing derivatives with differential polynomials and about partial sharing, using a toolkit that combines Zalcman–Pang rescaling with Nevanlinna theory. Collectively, these findings broaden the scope of normality criteria in complex analysis and supply a unified approach to partial sharing with differential polynomials.

Abstract

This article aims at finding sufficient conditions for a family of meromorphic functions to be normal by involving partial sharing of sets with differential polynomials. Moreover, corresponding results for normal meromorphic functions are also established which improve and generalize many known results.
Paper Structure (4 sections, 18 theorems, 99 equations)

This paper contains 4 sections, 18 theorems, 99 equations.

Table of Contents

  1. Introduction and Statement of Results
  2. Auxiliary Results
  3. Proofs of Main Results
  4. Statements and Declaration

Key Result

Theorem 1

Let $\mathcal{F}\subset\mathcal{M}(D)$ and $S=\left\{a_1, a_2, a_3\right\}$ be a set in $\mathbb{C}.$ If for each $f\in\mathcal{F},$$f(z)\in S\Leftrightarrow f'(z)\in ,$ then $\mathcal{F}$ is normal in $D.$

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Example 1
  • Remark 1
  • Example 2
  • Theorem 7
  • ...and 22 more