Sharp bounds of the Landau constants
Sharp bounds of the Landau constants
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DOI：
10.1090/s0025-5718-2010-02428-7
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发表时间：
2011-05
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通讯作者：
C. Mortici
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作者：
C. Mortici
本文的目的是建立朗道常数的新界限。 1. 介绍和动机 所有正整数 n 的朗道常数定义为 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 ；在复分析和傅里叶级数理论中的一些极值问题中发挥着重要作用。更准确地说，1913 年 Landau [6] 证明了 Gn 是表达式 j Pn k=1 akj 的最大值；对于 f (z) = P1 k=0 akz k 形式的所有函数，该函数在单位圆盘中解析且满足 es jf (z)j < 1；对于每个 jzj < 1：因此，Landau 常数的近似问题引起了许多作者的注意。特别是，Landau 本人研究了 Gn 的渐近行为，并证明了 Gn 1 lnn;然后 Watson [7] 建立了以下渐近公式 Gn = c0 + 1 ln (n+ 1) 1 4 (n+ 1) +O 1 n2 (n!1)： 这里，以及接下来的，c0 = 1 ( + 4 ln 2) = 1:06627:::; and = 0:577215::: 是欧拉-马斯切罗尼常数。 Brutman [3] 和 Falaleev [5] 的作品继续讨论朗道常数的近似问题，他们证明了对于每个非负整数 n； 1 + 1 ln (n+ 1) < Gn < 1:0663 + 1 ln (n+ 1) ;分别 1:0662 + 1 ln n+ 3 4 < Gn < 1:0916 + 1 ln n+ 3 4 ：我们通过以下方式改进上限，这也表明常数 3=4 是最好的可能。日期：2009年11月18日。1991年数学学科分类。初级 41A60；中学 26D15。
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.