Remarks on a Convexity Theorem of Raşa

In 2018 and 2019, Raşa presented two proofs for the log-convexity of Fn(x)=∑ν=0n(nνxν(1-x)n-ν)2,n=0,1,2,…,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F_n(x)=\sum _{\nu =0}^n \Bigl ( {n\atopwithdelims ()\nu } x^{\nu } (1-x)^{n-\nu } \Bigl )^2, \quad n=0,1,2, \ldots , \end{aligned}$$\end{document}on [0, 1]. Here, we offer a third proof of Raşa’s convexity result and we show that Fn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_n$$\end{document} is completely monotonic on [0, 1/2] and absolutely monotonic on [1/2, 1].