Let F be a distribution function supported on R+, which for the moment I'm prepared to assume to be continuous and strictly increasing. Let F^-1 be its right-continuous inverse:

F^-1(x)=inf{y:F(y)>x}. In particular F^-1(1)=infty. Since F is continuous and strictly increasing, F^-1 is also continuous and strictly increasing on (0,1) and is the true inverse of F in the usual sense.

I am interested in the following transformation of F: for fixed r>0, let F_r(x)=F(rF^-1(x)). Since F^-1(1)=infty, F_r is a proper distribution function on [0,1]. Also, note that the inverse of F_r is F_{1/r}.

Does anyone know if such a transformation has appeared anywhere in the literature already? The specific question I'm interested in is: if F and G are 2 distribution functions,

under what assumptions on F and G is it true that the mean of F_r is greater than or equal to the mean of G_r for r