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 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 Raşaʹs conjecture.