On a conjecture of Lamkin and Tkocz: Log-convexity of moments of Bernoulli sample means

Let $X_1,X_2,\dots$ be independent $\mathrm{B}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{o}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{i}(\theta )$ random variables, and let ${\bar{X}}_n=n^{-1}(X_1+\cdots +X_n)$. We prove that, for every real $p\ge 1$, the sequence $\{\mathsf{E}({\bar{X}}_n^p){\}}_{n\ge 1}$ is log-convex. This settles the Bernoulli case of a conjecture of Lamkin and Tkocz [Canad. Math. Bull., 65(2):271-278, 2022]. The proof conditions on the total number of successes among $2n$ trials and reduces the desired inequality to a convex-order comparison for a normalized quadratic function of a hypergeometric random variable. The log-convexity inequality is strict for $p>1$ and $0<\theta <1$.