I am trying to construct an example of a function $F(x)=\frac{\Phi_1^{-1}(x)}{\Phi_2^{-1}(x)}$, where $\Phi_1, \Phi_2$, $x\in R_+$ are Young's functions, so that the least concave majornat of $F$ would be increasing and concave.
So far, I have taken $\Phi_1^{-1}(x)=x^{1-1/q}\ln(c/t)$, $c>0$ and $\Phi_2=x^{1/p}$, where $1/p+1/q=1$, $1<q,p< \infty$. However, it seems, I am getting opposite-$F$ is decreasing and convex.