On the limits of real-valued functions in sets involving $ \psi $-density, and applications

TL;DR

This paper advances the understanding of limits of real-valued functions using $$-densities, improving growth results for functions on sets with positive density and introducing a new limit concept relevant to analysis and meromorphic functions.

Contribution

It introduces new limit results based on $$-densities and proposes a novel limit concept for real-valued functions, extending existing theories.

Findings

Improved bounds on the growth of non-decreasing functions in sets of positive density.

New limit concept for real-valued functions based on $$-densities.

Applications to real analysis and meromorphic function theory.

Abstract

We prove new results on upper and lower limits of real-valued functions by means of $\psi$-densities introduced by P. D. Barry in 1962. This allows us to improve several existing results on the growth of non-decreasing and unbounded real-valued functions in sets of positive density. The $\psi$-densities are also used to introduce a new concept of a limit for real-valued functions. The results in this paper are of interest in real analysis as well as in the theory of meromorphic functions.