The distance between maximum modulus points and the zero set of an entire function

Abstract

We obtain asymptotical bounds from below for the distance between a maximum modulus point and the zero set of an entire function of finite order. Known bounds(Macintyre,1938) are more precise but they are valid only for some maximum modulus points and can not be valid for all of them. Our bounds are valid for each maximum modulus point and are precise in some sense. Our method is based on Macintyre's ideas and on the techniques of proxi mate and strong proximate orders. Keywords: Entire function, Maximum modulus point, Zero set, Proximate order, Strong proximate order.