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(mp+(r,f)) where mp+(r,f) is the Lp norm of
log+|f(rei𝜃)| 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.