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

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 Pn′(x)/Pn(x) for x≥1, where Pn is the n-th Legendre polynomial. Here, we prove both upper and lower pointwise estimates for the ratios (Pn(λ)(x))′/Pn(λ)(x), x≥1, where Pn(λ) is the n-th ultraspherical polynomial. In particular, we validate Raşa's conjecture.