A lemma on the difference quotients

TL;DR

This paper extends the difference analogue of the lemma on the logarithmic derivative to certain meromorphic functions using a new growth lemma, analyzing exceptional sets and applying results to discrete equations.

Contribution

It introduces a new Borel type growth lemma and extends existing lemmas to broader classes of meromorphic functions, including discrete equations.

Findings

Extension of the lemma on difference quotients with finite logarithmic measure sets

Necessity of infinite linear measure for certain exceptional sets

Construction of entire functions where the difference quotient is not small

Abstract

Using a new Borel type growth lemma, we extend the difference analogue of the lemma on the logarithmic derivative due to Halburd and Korhonen to the case of meromorphic functions $f(z)$ such that $\log T(r,f)\leq r/(\log r)^{2+\nu}$, $\nu>0$, for all sufficiently large $r$. The method by Halburd and Korhonen implies an estimate for the lemma on difference quotients, where the exceptional set is of finite logarithmic measure. We show the necessity of this set by proving that it must be of infinite linear measure for meromorphic functions whose deficiency is dependent on the choice of the origin. In addition, we show that there is an infinite sequence of $r$ in the set for which $m(r,f(z+c)/f(z))$ is not small compared to $T(r,f)$ for entire functions constructed by Miles. We also give a discrete version of Borel type growth lemma and use it to extend Halburd's result on first order discrete equations of Malmuist type.