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${z:|z|<1}, hin H$ and$Omega_f(r)= sum_{n=0}^{infty} |a_n| r^n$. If$$eta_{fh}=varliminflimits_{ro1}frac{lnlnOmega_f(r)}{ln h(r)}=+infty,$$then Wiman's inequality$M_f(r)leq mu_f(r) ln^{1/2+delta}mu_f(r)$is true for all $rin (r_0, 1)ackslash E(delta)$, where $h-mbox{meas} E<+infty.$