Abstract
The aim of this paper is to establish new bounds of the Landau constants. 1. Introduction and Motivation The Landau constants de ned for all positive integers n by Gn = 1 + 1 2 2 + 1 3 2 4 2 + 1 3 5 2 4 6 2 + :::+ 1 3 ::: (2n 1) 2 4 ::: (2n) 2 ; play an important role in some extremal problems in complex analysis and in the theory of Fourier series. More precisely, in 1913 Landau [6] proved that Gn is the maximum of the expression j Pn k=1 akj ; with respect to all functions of the form f (z) = P1 k=0 akz k which is analytic in the unit disk and satis es jf (z)j < 1; for every jzj < 1: In consequence, the problem of approximation of the Landau constants have attracted the attention of many authors. In particular, Landau himself studied the asymptotic behaviour of Gn and showed that Gn 1 lnn; then Watson [7] established the following asymptotic formula Gn = c0 + 1 ln (n+ 1) 1 4 (n+ 1) +O 1 n2 (n!1): Here, and in what follows, c0 = 1 ( + 4 ln 2) = 1:06627:::; and = 0:577215::: is the Euler-Mascheroni constant. This problem of approximation of the Landau constants was continued in the works of Brutman [3] and Falaleev [5], who proved that for every non-negative integer n; 1 + 1 ln (n+ 1) < Gn < 1:0663 + 1 ln (n+ 1) ; respective 1:0662 + 1 ln n+ 3 4 < Gn < 1:0916 + 1 ln n+ 3 4 : We improve the upper bound in the following way, that also shows that the constant 3=4 is the best possible. Date : November 18, 2009. 1991 Mathematics Subject Classi cation. Primary 41A60; Secondary 26D15.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.