On the log-convexity of a Bernstein-like polynomials sequence

We prove that the sequence of the sum of the squares of the Bernstein polynomials is pointwise log-convex. There are given two proofs of this result: one by relating our sequence to the Legendre polynomials sequence and one by induction. I know of this problem from Professor Ioan Rasa, Cluj-Napoca. This work was presented at the International Conference on Approximation Theory and its Applications, Sibiu, 2022, dedicated to the scientific work of Professor Ioan Rasa on the occasion of his 70th anniversary.