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 tixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304644&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304644&_rdoc=1&_issn=0022247X&md5=17f4ab08da59c60511147b497722729d" title="Click to view the MathML source">R(a)−B(a) at tixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304644&_mathId=si2.gif&_user=111111111&_pii=S0022247X16304644&_rdoc=1&_issn=0022247X&md5=4636e89e16295641d75436b88e9c8d10" title="Click to view the MathML source">a=0 and at tixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304644&_mathId=si3.gif&_user=111111111&_pii=S0022247X16304644&_rdoc=1&_issn=0022247X&md5=0f2818324b42a3495644c958f73dc622" title="Click to view the MathML source">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 tixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304644&_mathId=si4.gif&_user=111111111&_pii=S0022247X16304644&_rdoc=1&_issn=0022247X&md5=2882654c78543459a9138348228e9163" title="Click to view the MathML source">R(a) and the beta function tixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304644&_mathId=si5.gif&_user=111111111&_pii=S0022247X16304644&_rdoc=1&_issn=0022247X&md5=6f77e6f44e853cc14f1fcd32b18141bb" title="Click to view the MathML source">B(a)≡B(a,1−a), and obtain asymptotically sharp lower and upper bounds for tixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304644&_mathId=si4.gif&_user=111111111&_pii=S0022247X16304644&_rdoc=1&_issn=0022247X&md5=2882654c78543459a9138348228e9163" title="Click to view the MathML source">R(a) in terms of tixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304644&_mathId=si6.gif&_user=111111111&_pii=S0022247X16304644&_rdoc=1&_issn=0022247X&md5=f6af48cb501693bb9f63dc891dedf525" title="Click to view the MathML source">B(a) and polynomials. In addition, some properties of the Riemann zeta function tixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304644&_mathId=si7.gif&_user=111111111&_pii=S0022247X16304644&_rdoc=1&_issn=0022247X&md5=24afbaaa7b770a55f73809396f67c02d" title="Click to view the MathML source">ζ(n), tixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304644&_mathId=si8.gif&_user=111111111&_pii=S0022247X16304644&_rdoc=1&_issn=0022247X&md5=554b913bffeb1fefd55b83088e1f5d52" title="Click to view the MathML source">n∈N, and its related sums are derived.