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

TL;DR

This paper develops a second main theorem for meromorphic functions with finite growth index on a complex disc, and extends Nevanlinna's five values theorem for functions sharing five small functions.

Contribution

It introduces a second main theorem with detailed estimates and generalizes the five values theorem for meromorphic functions sharing small functions.

Findings

Established a second main theorem with truncated counting functions.

Generalized Nevanlinna's five values theorem for small functions.

Provided improved estimates for small term contributions.

Abstract

This paper has twofold. The first is to establish a second main theorem for meromorphic functions on the complex disc $\Delta (R_0)\subset\mathbb C$ 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 $\Delta (R_0)$.