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

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 $P_n^{\prime }(x)/P_n(x)$ for $x\ge 1$, where $P_n$ is the $n$-th Legendre polynomial. Here, we prove both upper and lower pointwise estimates for the ratios $(P_n^{(\lambda )}(x){)}^{\prime }/P_n^{(\lambda )}(x)$, $x\ge 1$, where $P_n^{(\lambda )}$ is the $n$-th ultraspherical polynomial. In particular, we validate Raşa's conjecture.