Browsing by Subject "Entire function"

Now showing 1 - 7 of 7

Open Access
Distance between a maximum modulus point and zero set of an entire function
(Taylor & Francis, 2003-07) Ostrovskii, I.; Üreyen, A. E.
Let f be an entire function of finite positive order. A maximum modulus point is a point w such that j fðwÞj maxfjfðzÞj: jzjjwjg. We obtain lower bounds for the distance between a maximum modulus point w and the zero set of f. These bounds are valid for all sufficiently large values of jwj.
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.
Open Access
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.
Open Access
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.
Open Access
On power series with one singular point at circumference of convergence
(2002) Yardımcı, Umut
We obtain two refinements of Faber’s theorem related to power series with one singular point at circumference of convergence. The first one characterizes growth at the singular point in more precise scale of growth. The second one characterizes the growth at the singular point using series expansions both inside and outside the disc of convergence.
Open Access
On the zero distributionof remainders of entire power series
(Taylor & Francis, 2001) Ostrovskll, I. V.
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.
Open Access
On the zeros of tails of power series
(Springer, 2000) Ostrovskii, Iossif Vladimirovich; Havin, V. P.; Nikolski, N. K.
Let f(z)=∑k=0∞akzkf(z)=∑k=0∞akzk (1.1) be a power series with a positive radius of convergence. Let sn(z)=∑k=0nakzk,tn(z)=∑k=n+1∞akzksn(z)=∑k=0nakzk,tn(z)=∑k=n+1∞akzk be its nth section and nth tail, respectively.