Inequalities for the Chi Square Distribution Function

Let Pa x.s 1rG x..H0a eyttxy1dt be the chi square distribution function, and let Mt u,¨;a.be the weighted power mean of order t. We prove the following: If x,y)0 x/y.and a g 0, 1.are fixed.real numbers, then the inequalities Mr Pa x.,Pa y.;a.-Pa axq 1ya.y.-Ms Pa x.,Pa y.;a. are valid for all real numbers a)0 if and only if rF0 and ss`. In particular, we obtain that for all a)0 the function x¬Pa x. is log-concave on 0, `.. This proves a conjecture of M. Merkle