Borel lemma: geometric progression and zeta-functions
Qi Han, Jingbo Liu, Nadeem Malik
TL;DR
The paper advances refined Borel-type inequalities for meromorphic functions of infinite order by weaving together Borel, Nevanlinna, and Hayman formulations through a unified $s>1$ parameter and by employing zeta-function-based exceptional-set measures, including the Riemann and Hurwitz variants. It analyzes the impact of shifts $R=r+1/T^s(r)$ or $R=r+1/T(r)$ on bounds for $T(R,f)$, showing that the dominant term remains $2\log^+T(r,f)$ while the exceptional-set sizes are governed by $\gamma(s)$, $\zeta(s)$, and $\zeta(s,a)$, with explicit sharpness considerations at $s=2$. The work demonstrates that Nevanlinna’s bound yields the smallest exceptional set among the three, Hayman’s is larger, and Borel’s is largest, while also extending Fernández Árias’ refinement to a scenario with an exceptional set $S_e$ independent of a parameter $\sigma\in(0,1)$ and providing concrete constants (e.g., $\zeta(2)=\pi^2/6$) and fast-growth examples. Collectively, these results yield sharper growth-inequality statements and more precise control over exceptional sets in value-distribution theory for fast-growing meromorphic functions, with potential implications for the Second Main Theorem and its applications.
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 $ζ\bigl(2,\sqrt{2}+1\bigr)\leq0.52<2$ for the Hurwitz zeta-function $ζ(s,a)$. 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 Fernández Árias \cite{aFA1}.