On entire functions of infinite order with radially distributed zeros

Joseph B. Miles

Vol. 81 (1979), No. 1, 131–157

DOI: 10.2140/pjm.1979.81.131

Abstract

Suppose f is an entire function of infinite order with zeros restricted to a finite number of rays through the origin. It is shown for p > 1 that N(r,0) = o(m_p⁺(r,f)) where m_p⁺(r,f) is the L^p norm of log ⁺|f(re^i𝜃)| and in addition that N(r,0) = o(T(r,f)) as r tends to infinity omitting values in an exceptional set E of zero logarithmic density. The set E is shown by example in general to be nonempty, even for functions with zeros on a single ray and arbitrarily slow infinite rate of growth. These results settle certain questions arising from previous work of Edrei, Fuchs, and Hellerstein and of Hellerstein and Shea.

Mathematical Subject Classification 2000

Primary: 30D35

Milestones

Received: 1 February 1978

Published: 1 March 1979