  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 $\int_0^\infty \psi(x) dx
<\infty$.
