Remarks on a Convexity Theorem of Raşa

Abstract

In 2018 and 2019, Raşa presented two proofs for the log-convexity of

$$\begin{aligned} F_n(x)=\sum _{\nu =0}^n \Bigl ( {n\atopwithdelims ()\nu } x^{\nu } (1-x)^{n-\nu } \Bigl )^2, \quad n=0,1,2, \ldots , \end{aligned}$$

on [0, 1]. Here, we offer a third proof of Raşa’s convexity result and we show that $F_n$ is completely monotonic on [0, 1/2] and absolutely monotonic on [1/2, 1].