Maximum modulus points and zero sets of entire functions of regular growth

Abstract

We obtain lower asymptotic at ∞ estimates of the distance between a maximum modulus point and zero set of an entire function provided that the function is of regular growth with respect to a proximate order. The more regular the growth is the better the estimates are, and they are sharp in some sense. The case of infinite order is also considered; in this case a suitable analogue of usual proximate order is exploited. © 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.