Some properties of the difference between the Ramanujan constant and beta function

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• 作者：Song-Liang Qiu ; ^{sl_qiu@zstu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Xiao-Yan Ma ; Ti-Ren Huang
• 关键词：The Ramanujan constant ; Beta function ; Monotonicity and convexity ; Functional inequalities ; Power series ; The Riemann zeta function
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：114-129
• 全文大小：368 K

文摘

The authors present the power series expansions of the function R(a)−B(a)$R(a)-B(a)$ at a=0$a=0$ and at a=1/2$a=1/2$, show the monotonicity and convexity properties of certain familiar combinations defined in terms of polynomials and the difference between the so-called Ramanujan constant R(a)$R(a)$ and the beta function B(a)≡B(a,1−a)$B(a)\equiv B(a,1-a)$, and obtain asymptotically sharp lower and upper bounds for R(a)$R(a)$ in terms of B(a)$B(a)$ and polynomials. In addition, some properties of the Riemann zeta function ζ(n)$\zeta (n)$, n∈N$n\in \mathbb{N}$, and its related sums are derived.