Browsing by Subject "Zero set"

Now showing 1 - 3 of 3

Open Access
Distance between a maximum point and the zero set of an entire function
(2006) Üreyen, Adem Ersin
We obtain asymptotical bounds from below for the distance between a maximum modulus point and the zero set of an entire function. Known bounds (Macintyre, 1938) are more precise, but they are valid only for some maximum modulus points. Our bounds are valid for all maximum modulus points and moreover, up to a constant factor, they are unimprovable. We consider entire functions of regular growth and obtain better bounds for these functions. We separately study the functions which have very slow growth. We show that the growth of these functions can not be very regular and obtain precise bounds for their growth irregularity. Our bounds are expressed in terms of some smooth majorants of the growth function. These majorants are defined by using orders, types, (strong) proximate orders of entire functions.
Unknown
The distance between maximum modulus points and the zero set of an entire function
(2001) Üreyen, Adem Ersin
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.
Unknown
On maximum modulus points and zero sets of entire functions of regular growth
(2008) Ostrovskii I.; Üreyen, A.E.
Let f be an entire function. We denote by R(w, f) the distance between a maximum modulus point w and the zero set of f. In a previous paper, the authors obtained asymptotical lower bounds for R(w, f) as |w| → ∞ for functions of finite positive order and regular growth. In this work we extend those results to functions of either zero or infinite order and show that our results are sharp in sense of order. Copyright © 2008 Rocky Mountain Mathematics Consortium.