Proof of a conjecture of Granath on optimal bounds of the Landau constants

JOURNAL OF APPROXIMATION THEORY

卷:204

Article

Zhao, Chun-Ru¹  Long, Wen-Gao¹  Zhao, Yu-Qiu¹ 

[1] Sun Yat Sen Univ, Dept Math, Guangzhou 510275, Guangdong, Peoples R China

关键词: Landau constants;    Second-order linear difference equation;    Sharper bound;    Asymptotic expansion;    Hypergeometric function;   

DOI  :  10.1016/j.jat.2015.12.004

来源: Elsevier

【 摘 要 】

We study the asymptotic expansion for the Landau constants G(n), pi G(n) similar to ln(16N) + gamma + Sigma(infinity)(k=1) alpha(k)/N-k as n -> infinity, where N = n+1, and gamma is Euler's constant. We show that the signs of the coefficients alpha(k) demonstrate a periodic behavior such that (-1)(l(l+1)/2) alpha(l+1) < 0 for all l. We further prove a conjecture of Granath which states that (-1)(l(l+1)/2) epsilon(l)(N) < 0 for l = 0, 1, 2,... and n = 0, 1, 2,..., epsilon(l)(N) being the error due to truncation at the lth order term. Consequently, we also obtain the sharp bounds up to arbitrary orders of the form ln(16N) + gamma + Sigma(p)(k=1) alpha(k)/N-k < pi G(n) < ln(16N) + gamma + Sigma(q)(k=1) alpha(k)/N-k for all n = 0, 1, 2,..., all p = 4s+1, 4s+2 and q = 4m, 4m+3, with s = 0,1, 2,... and m = 0,1, 2,... (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free