STOLARSKYS INEQUALITY WITH GENERAL WEIGHTS

Recently Stolarsky proved that the inquality () integral(0)(1) g(x(1/ (A+b))) dx greater than or equal to integral(0)(1) g(x(1/a)) dx integr al(0)(1) g(x(1/b) dx holds for every a, b > 0 and every nonincreasing function on [O, 1] satisfying 0 less than or equal to g(u) less than o r equal to 1. In this paper we prove a weighted version of this inequa lity. Our proof is based on a generalized Chebyshev inequality. In par ticular, our result shows that the inequality () holds for every func tion g of bounded variation. We also generalize another inequality by Stolarsky concerning the Gamma-function.