JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS Vol. 123, No. 2, May 1, 1987.

Sharper Wiman Inequality for Entire Functions
with Rapidly Oscillating Coefficients

J. MICHAEL STEELE*

Department of Statistics, E-220 Engineering Quadrangle,
Princeton University, Prineeton, New Jersey 08544

Submitted by R. P. Boas

For entire functions of the form $\sum_{n=0}^\infty a_n \exp(i \theta_n t) z^n$ where the $\theta_n$ are integers satisfying
the Hadamard gap condition it is proved that Wiman's inequality can be improved
to $M(r) \leq \mu (r) (\log \mu (r))^(1/4) (\log \log \mu (r)) ^{1/2+ \delta} for almost every t and all r except a set$E_\delta (t)\$ of finite logarithmic measure.

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