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Second main theorem and uniqueness problem of meromorphic functions with finite growth index sharing five small functions on a complex disc

Si Duc Quang

Abstract

This paper has twofold. The first is to establish a second main theorem for meromorphic functions on the complex disc $Δ(R_0)\subset\mathbb C$ with finite growth index and small functions, where the counting functions are truncated to level $1$ and the small term is more detailed estimated. The second is to prove a generalization and improvement of the five values theorem of Nevanlinna for the case of five small functions on the complex disc $Δ(R_0)$.

Second main theorem and uniqueness problem of meromorphic functions with finite growth index sharing five small functions on a complex disc

Abstract

This paper has twofold. The first is to establish a second main theorem for meromorphic functions on the complex disc with finite growth index and small functions, where the counting functions are truncated to level and the small term is more detailed estimated. The second is to prove a generalization and improvement of the five values theorem of Nevanlinna for the case of five small functions on the complex disc .
Paper Structure (3 sections, 5 theorems, 69 equations)

This paper contains 3 sections, 5 theorems, 69 equations.

Table of Contents

  1. Introduction and Results
  2. Truncated second main theorem for meromorphic functions and small functions on the complex disc
  3. Uniqueness theorem for meromorphic functions with finite growth index sharing five small functions

Key Result

Theorem 1.1

Let $f$ be a non-constant meromorphic function on $\Delta (R_0)$ and let $a_1,\ldots,a_5$ be five distinct small functions (with respect to $f$). Let $\gamma(r)$ be a non-negative measurable function defined on $(0,R_0)$ with $\int_0^{R_0}\gamma (r)dr=+\infty$. Then, for every $\varepsilon >0$,

Theorems & Definitions (10)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1: Lemma on logarithmic derivative RS
  • Theorem 2.4: see RS
  • proof : Proof of Theorem \ref{['1.1']}
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['1.2']}