On the log-concavity of $n$-th root of a sequence

TL;DR

This paper confirms Sun's conjecture that certain sequences involving binomial coefficients are log-concave when taking the n-th root, using semi-automatic and analytic methods.

Contribution

It introduces two methods, semi-automatic and analytic, to prove the log-concavity of specific sequences involving binomial coefficients, confirming a recent conjecture.

Findings

Confirmed Sun's conjecture on log-concavity of the sequences

Developed a criterion and used Mathematica for semi-automatic proof

Applied Xia's result for the analytic proof

Abstract

In recent years, the log-concavity of $\{\sqrt[n]{S_n}\}_{n\geq 1}$ have been received a lot of attention. Very recently, Sun posed the following conjecture in his new book: the sequences $\{\sqrt[n]{a_n}\}_{n\geq 2}$ and $\{ \sqrt[n]{b_n}\}_{n\geq 1}$ are log-concave, where \[ a_n:= \frac{1}{n}\sum_{k=0}^{n-1} \frac{{n-1\choose k}^2{n+k\choose k}^2 }{4k^2-1} \] and \[ b_n:= \frac{1}{n^3}\sum_{k=0}^{n-1} (3k^2+3k+1){n-1\choose k}^2 {n+k\choose k}^2. \] 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 $\{\sqrt[n]{S_n}\}_{n\geq 1}$ given by us and a mathematica package due to Hou and Zhang, while the analytic method relies on a result due to Xia.