Sharp bounds of the Landau constants

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.