Inequalities for ultraspherical polynomials. Proof of a conjecture of I. Ra\c{s}a

A recent conjecture by I. Ra\c{s}a 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 $P_n^{\mathrm{\prime }}(x)\mathrm{/}{P}_n(x)$ 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 $(P_n^{(\lambda )}(x){)}^{\mathrm{\prime }}\mathrm{/}{P}_n^{(\lambda )}(x)$, $x\ge 1$, where $P_n^{(\lambda )}$ is the $n$-th ultraspherical polynomial. In particular, we validate Ra\c{s}a’s conjecture.