A description of the algebras of analytic functions admitting localization of ideals

Let$\mathbf D=\{z\in\mathbf C:|z|<1\}$ and let $A_\varphi(\mathbf D)$ be the algebra of all analytic functions $f$ in $\mathbf D$
for which $\log|f(z)|\leqslant C_f\varphi\biggl(\dfrac{1}{1-|z|}\biggr)$, $z\in\mathbf D$. Under in known restrictions regarding the regularity of the growth of the function $\varphi$, one proves
THEOREM. In order that each closed ideal $I$, $I\subset A_\varphi(\mathbf D)$, be local, it is necessary and sufficient that one should have
$$ \int_1^\infty\biggl(\dfrac{\varphi(x)}{x^3}\biggr)^{1/2}dx=\infty. $$
be the algebra of all analytic functions.
Here, the localness of the ideal $I$ means that $I=\{f\in A_\varphi(\mathbf D):k_f\geqslant k_I\}$, where $k_f(\zeta)$ is the multiplicity of a zero of the function $f$ at the point $\zeta$, $k_I(\zeta)=\min_{f\in I}k_f(\zeta)$.