On the log-concavity of the n-th root of sequences

Abstract

In recent years, the log-concavity of the n-th root of a sequence ${\left\{\sqrt[n]{S_n}\right\}}_{n\ge 1}$ has been received a lot of attention. Recently, Sun posed the following conjecture in his new book: the sequences ${\left\{\sqrt[n]{a_n}\right\}}_{n\ge 2}$ and ${\left\{\sqrt[n]{b_n}\right\}}_{n\ge 1}$ are log-concave, where$a_n:=\frac{1}{n}\sum _{k=0}^{n-1}\frac{{\left(\begin{array}{c}n-1\\ k\end{array}\right)}^2{\left(\begin{array}{c}n+k\\ k\end{array}\right)}^2}{4k^2-1}$ and$b_n:=\frac{1}{n^3}\sum _{k=0}^{n-1}(3k^2+3k+1){\left(\begin{array}{c}n-1\\ k\end{array}\right)}^2{\left(\begin{array}{c}n+k\\ k\end{array}\right)}^2.$ In this paper, two methods, semi-automatic and analytic methods, are used to confirm Sun's conjecture. The semi-automatic method relies on a criterion on the log-concavity of ${\left\{\sqrt[n]{S_n}\right\}}_{n\ge 1}$ given by us and a mathematica package due to Hou and Zhang, while the analytic method relies on a result due to Xia.