Direct analogues of Wiman's inequality for analytic functions in the unit disc

Abstract

Let $f(z)=\sum_{n=0}^{\infty} a_n z^n$ be an analytic function on $\{z:|z|<1\},\ h\in H$ and $\Omega_f(r)= \sum_{n=0}^{\infty} |a_n| r^n$. If$$\beta_{fh}=\liminf\limits_{r\to1}\frac{\ln\ln\Omega_f(r)}{\ln h(r)}=+\infty,$$then Wiman's inequality $M_f(r)\leq \mu_f(r) \ln^{1/2+\delta}\mu_f(r)$ is true for all $r\in (r_0, 1)\backslash E(\delta)$, where $h-\mbox{meas}\ E<+\infty.$