On the zero distributionof remainders of entire power series

Abstract

It has been shown by the author that, if all remainders of the power series of an entire function f have only real positive zeros, then log M(r, f) = O((1og r)'), r -+ ca. The main results of the paper are the following: (i) if at least two different remainders have only real positive zeros, then logM(r,f) = O(fi, r+ ca; (ii) this estimate cannot be improved even in the case if one replaces two by any given finite number of remainders.