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 $T(r)\geq1$ satisfies $T\bigl(r+\frac{1}{T(r)}\bigr)<2T(r)$ outside a possible exceptional set of linear measure $2$. We note in this work $T(r)$ satisfies a sharper inequality $T\bigl(r+\frac{1}{T(r)}\bigr)<\bigl(\sqrt{T(r)}+1\bigr)^2\leq2T(r)$, if $T(r)\geq\bigl(\sqrt{2}+1\bigr)^2$, outside a possible exceptional set of linear measure $\zeta\bigl(2,\sqrt{2}+1\bigr)\leq0.52<2$ for the Hurwitz zeta-function $\zeta(s,a)$. This result is worth noting, provided the set of $r$ in which $1\leq T(r)<\bigl(\sqrt{2}+1\bigr)^2$ 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 Fern\'{a}ndez \'{A}rias \cite{aFA1}.