Characteristic, maximum modulus and value distribution

Hayman, W. K.; Rossi, J. F.

doi:10.1090/S0002-9947-1984-0743737-2

Characteristic, maximum modulus and value distribution
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by W. K. Hayman and J. F. Rossi PDF

Trans. Amer. Math. Soc. 284 (1984), 651-664 Request permission

Abstract:

Let $f$ be an entire function such that $\log M(r,f)\sim T(r,f)$ on a set $E$ of positive upper density. Then $f$ has no finite deficient values. In fact, if we assume that $E$ has density one and $f$ has nonzero order, then the roots of all equations $f(z) = a$ are equidistributed in angles. In view of a recent result of Murai [6] the conclusions hold in particular for entire functions with Fejér gaps.

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