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On the two conjectures

Nabadwip Sarkar, Debabrata Pramanik, Lata Mahato

TL;DR

The paper addresses the uniqueness problem for transcendental entire functions that share a function with their higher-order difference operators, extending Brück-type conjectures B and C by removing the $ ho(f)<2$ constraint. Using Nevanlinna theory for difference operators, Hadamard factorization, and a suite of growth- and difference-operator lemmas, the authors prove that if $f$ and $\Delta_c^n f$ (or $f-\alpha$ and $\Delta_c^n f-\alpha$) share the value $0$ CM under suitable small-function conditions, then $f$ must be of exponential form $f(z)=c_0e^{dz}$ (or $f(z)=\alpha(z)+H(z)e^{dz}$ with explicit relations among the components). They provide sharp examples to demonstrate the necessity of the imposed conditions. These results confirm the conjectures of Liu and Laine and of Zhang et al., and advance the understanding of value-sharing phenomena for difference operators in complex analysis with potential implications for discrete analogues of differential equations.

Abstract

In this paper, we investigate the uniqueness problem of entire functions that share an entire function with their higher-order difference operators. We obtain two results that confirm the conjectures posed by Liu and Laine \cite{LL1} and by Zhang et al. \cite{ZKL1}, respectively. In addition, we present several relevant examples to further illustrate and support our findings

On the two conjectures

TL;DR

The paper addresses the uniqueness problem for transcendental entire functions that share a function with their higher-order difference operators, extending Brück-type conjectures B and C by removing the constraint. Using Nevanlinna theory for difference operators, Hadamard factorization, and a suite of growth- and difference-operator lemmas, the authors prove that if and (or and ) share the value CM under suitable small-function conditions, then must be of exponential form (or with explicit relations among the components). They provide sharp examples to demonstrate the necessity of the imposed conditions. These results confirm the conjectures of Liu and Laine and of Zhang et al., and advance the understanding of value-sharing phenomena for difference operators in complex analysis with potential implications for discrete analogues of differential equations.

Abstract

In this paper, we investigate the uniqueness problem of entire functions that share an entire function with their higher-order difference operators. We obtain two results that confirm the conjectures posed by Liu and Laine \cite{LL1} and by Zhang et al. \cite{ZKL1}, respectively. In addition, we present several relevant examples to further illustrate and support our findings

Paper Structure

This paper contains 6 sections, 7 theorems, 128 equations.

Table of Contents

  1. Introduction and main results
  2. In connection with Conjecture B
  3. In connection with Conjecture C
  4. Auxiliary lemmas
  5. Proofs of main results
  6. Statements and declarations

Key Result

Theorem 1.1

Let $f$ be a transcendental entire function such that $\rho(f)<+\infty$, $c\in\mathbb{C}\backslash \{0\}$ such that $\Delta_c^nf\not\equiv 0$ and $\lambda(f)<\rho(f)$. If $f$ and $\Delta^n_cf$ share $0$ CM, then $f$ must be of form $f(z)=c_0e^{dz}$, where $c_0, d\in\mathbb{C}\backslash \{0\}$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Example 1.1
  • Example 1.2
  • Theorem 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6
  • Lemma 2.1
  • Lemma 2.2
  • ...and 6 more