Jointly maximal products in weighted growth spaces

It is shown that for any non-decreasing, continuous and unbounded doubling function $\text{om}$ on $[0,1)$, there exist two analytic infinite products $f_0$ and $f_1$ such that the asymptotic relation $\mathrm{\mid }{f}_0(z)\mathrm{\mid }+\mathrm{\mid }{f}_1(z)\mathrm{\mid }\asymp \text{om}(\mathrm{\mid }z\mathrm{\mid })$ 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 $r\to 1^{-}$, and hence give examples of analytic functions for which the Nevanlinna characteristic admits the regular slow growth induced by $\omega$.