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

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