Inequalities Concerning Maximum Modulus and Zeros of Random Entire Functions

TL;DR

This paper investigates inequalities relating the maximum modulus and zeros of random entire functions, establishing bounds that hold almost surely and extending Nevanlinna theory to the random setting.

Contribution

It introduces a family of random entire functions and proves almost sure inequalities linking maximum modulus and zero counting functions, including a version of Nevanlinna's second main theorem for these functions.

Findings

Almost sure bounds on the difference between log maximum modulus and zero counting function.

Extension of Nevanlinna's second main theorem to random entire functions.

Bounded characteristic function by a weighted counting function for most random entire functions.

Abstract

Let $f_\omega(z)=\sum\limits_{j=0}^{\infty}\chi_j(\omega) a_j z^j$ be a random entire function, where $\chi_j(\omega)$ are independent and identically distributed random variables defined on a probability space $(\Omega, \mathcal{F}, \mu)$. In this paper, we first define a family of random entire functions, which includes Gaussian, Rademacher, Steinhaus entire functions. Then, we prove that, for almost all functions in the family and for any constant $C>1$, there exist a constant $r_0=r_0(\omega)$ and a set $E\subset [e, \infty)$ of finite logarithmic measure such that, for $r>r_0$ and $r\notin E$, $$ |\log M(r, f)- N(r,0, f_\omega)|\le (C/A)^{\frac1{B}}\log^{\frac1{B}}\log M(r,f) +\log\log M(r, f), \qquad a.s. $$ where $A, B$ are constants, $M(r, f)$ is the maximum modulus, and $N(r, 0, f)$ is the weighted counting-zero function of $f$. As a by-product of our main results, we prove Nevanlinna's second main theorem for random entire functions. Thus, the characteristic function of almost all functions in the family is bounded above by a weighed counting function, rather than by two weighted counting functions in the classical Nevanlinna theory. For instance, we show that, for almost all Gaussian entire functions $f_\omega$ and for any $\epsilon>0$, there is $r_0$ such that, for $r>r_0$, $$ T(r, f) \le N(r,0, f_\omega)+(\frac12+\epsilon) \log T(r, f). $$