Inequalities for ultraspherical polynomials. Proof of a conjecture of I. Raşa

2014/02/26 by Geno Nikolov, Nikolov, Geno
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA

paper · pdf · doi:10.48550/arxiv.1402.6539

arxiv created 2014/02/26 · arxiv updated 2014/02/27

Abstract

A recent conjecture by I. Raşa asserts that the sum of the squared Bernstein basis polynomials is a convex function in [0,1]. This conjecture turns out to be equivalent to a certain upper pointwise estimate of the ratio P_n^′(x)/P_n(x) for x≥ 1, where P_n is the n-th Legendre polynomial. Here, we prove both upper and lower pointwise estimates for the ratios (P_n^(λ)(x))^′/P_n^(λ)(x), ~x≥ 1, where P_n^(λ) is the n-th ultraspherical polynomial. In particular, we validate Raşa's conjecture.

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