Jointly maximal products in weighted growth spaces

TL;DR

The paper constructs two analytic functions in the unit disk whose combined magnitude and growth rate precisely match a given doubling function, illustrating examples of functions with controlled slow growth in Nevanlinna theory.

Contribution

It demonstrates the existence of two analytic infinite products that asymptotically match a specified doubling function's growth in the unit disk, providing new examples in growth space analysis.

Findings

Existence of functions matching the growth of any doubling function

Both functions have Nevanlinna characteristic asymptotic to log of the doubling function

Provides explicit examples of functions with slow regular growth

Abstract

It is shown that for any non-decreasing, continuous and unbounded doubling function $\om$ on $[0,1)$, there exist two analytic infinite products $f_0$ and $f_1$ such that the asymptotic relation $|f_0(z)| + |f_1(z)| \asymp \om(|z|)$ is satisfied for all $z$ in the unit disc. It is also shown that both functions $f_j$ for $j=0,1$ satisfy $T(r,f_j)\asymp\log\omega(r)$, as $r\to1^-$, and hence give examples of analytic functions for which the Nevanlinna characteristic admits the regular slow growth induced by $\omega$.