Inequalities for ultraspherical polynomials. Proof of a conjecture of I. Ra艧a

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• 作者：Geno Nikolov ^{geno@fmi.uni-sofia.bg" class="auth_mail}
• 关键词：Bernstein polynomials ; Legendre polynomials ; Ultraspherical polynomials
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：15 October 2014
• 年：2014
• 卷：418
• 期：2
• 页码：852-860
• 全文大小：243 K

文摘

A recent conjecture by I. Ra艧a asserts that the sum of the squared Bernstein basis polynomials is a convex function in X14003667&_mathId=si1.gif&_user=111111111&_pii=S0022247X14003667&_rdoc=1&_issn=0022247X&md5=3dc83f416876c9f298a3ced281dc0862" title="Click to view the MathML source">[0,1]$[0,1]$. This conjecture turns out to be equivalent to a certain upper pointwise estimate of the ratio mi>Pmi><mi>nmi>′(<mi>xmi>)/<mi>Pmi><mi>nmi>(<mi>xmi>) for x≥1mi>xmi>≥1, where P_nmi>Pmi><mi>nmi> is the n  -th Legendre polynomial. Here, we prove both upper and lower pointwise estimates for the ratios 142cc8f615c843">mi>Pmi><mi>nmi>(<mi>位mi>)(<mi>xmi>))′/<mi>Pmi><mi>nmi>(<mi>位mi>)(<mi>xmi>), x≥1mi>xmi>≥1, where mi>Pmi><mi>nmi>(<mi>位mi>) is the n-th ultraspherical polynomial. In particular, we validate Ra艧a's conjecture.