Sharper Wiman inequality for entire functions with rapidly oscillating coefficients (Q1096033)

Let \(f(z)=\sum^{\infty}_{k=0}a_ kz^ k\) represent an entire function and \(\{n_ k\}\) be a sequence of positive integers for which \(n_{k+1}/n_ k\geq q>1\). For a real number t define \[ g(z,t)=\sum^{\infty}_{k=0}(a_ ke^{itn}k)(z^ k) \] and \[ M(r,t)=\max | g(z,t)| \quad (| z| =r). \] For \(r>0\), let \(\mu (r)=\max_{k}| a_ k| r^ k\). Then for any \(\delta >0\) and almost every t, \[ M(r,t)\leq \mu (r)(\log \mu (r))^{1/4}(\log \log \mu (r))^{1+\delta} \] for all \(r>0\) except for a set \(E_{\delta}(t)\) of finite logarithmic measure. A slightly strongler result is proved - one analogous to a theorem of \textit{P. Erdős} and \textit{A. Rényi} [Zastos. Mat. 10, 47-55 (1969)]. Ingredients in the new result are parallel arguments to those of Erdős and Rényi and some inequalities of P. C. Rosenbloom. An example in the paper under review shows that the 1/4 in the result cannot be replaced by 1/4-\(\epsilon\) for any \(\epsilon >0\).