Borel lemma: geometric progression and zeta-functions

Abstract

In the proof of the classical Borel lemma \cite{eB} by Hayman \cite{wkH}, each continuous increasing function $T(r)\ge 1$ satisfies $T(r+\frac{1}{T(r)})<2T(r)$ outside a possible exceptional set of linear measure $2$. We note in this work $T(r)$ satisfies a sharper inequality $T(r+\frac{1}{T(r)})<(\sqrt{T(r)}+1{)}^2\le 2T(r)$, if $T(r)\ge (\sqrt{2}+1{)}^2$, outside a possible exceptional set of linear measure $\hat{I}\P (2,\sqrt{2}+1)\le 0.52<2$ for the Hurwitz zeta-function $\hat{I}\P (s,a)$. This result is worth noting, provided the set of $r$ in which $1\le T(r)<(\sqrt{2}+1{)}^2$ has linear measure less than $1.48$. Focusing exclusively on meromorphic functions of infinite order, we utilize Hinkkanen's Second Main Theorem \cite{aH}, draw comparisons with Borel \cite{eB}, Nevanlinna \cite{rN}, and Hayman \cite{wkH}, and finally generalize Fernández Árias \cite{aFA1}.