The problem of ideals of $H^\infty$: beyond the exponent 3/2

Abstract

The paper deals with the problem of ideals of $H^\infty$: describe increasing functions $ϕ\ge 0$ such that for all bounded analytic functions $f_1,f_2,...,f_n, τ$ in the unit disc $D$ the condition $|τ(z) | \le ϕ(\sum_k |f_k(z)|)$ for all $z\in D$, implies that $τ$ belong to the ideal generated by $f_1,f_2,...,f_n$. It was proved earlier by the author that the function $ϕ(s) =s^2$ does not work. The main result of the paper is that one can take for $ϕ$ any function of form $ϕ(s) =s^2 ψ(\ln s^{-2})$, where $ψ$ is a bounded non-increasing function on $[0, \infty)$ satisfying $\int_0^\infty ψ(x) dx <\infty$.