Stolarsky’s inequality with general weights

Abstract:

Recently Stolarsky proved that the inquality \begin{equation}\tag {$\ast $} \int _0^1 {g({x^{1/(a + b)}}) dx \geq \int _0^1 {g({x^{1/a}}) dx\int _0^1 {g({x^{1/b}}) dx} } } \end{equation} holds for every $a,b > 0$ and every nonincreasing function on [0, 1] satisfying $0 \leq g(u) \leq 1$. In this paper we prove a weighted version of this inequality. Our proof is based on a generalized Chebyshev inequality. In particular, our result shows that the inequality $( \ast )$ holds for every function g of bounded variation. We also generalize another inequality by Stolarsky concerning the $\Gamma$-function.