Derivative of Youngs function

Question

Let $\Phi_2^{-1}(t)=t^{1/p}$ and $\Phi^{-1}(t)=t^{1-1/q}[\log(t+t^{-1})]^{-1}$, $t\in R_+$, where $1<p,q,<\infty, 1/p+1/q>1$. Then, $\Phi_2, \Phi$ are Yong functions.

Consider $$ F(t)=\frac{\Phi_2^{-1}(t)}{\Phi^{-1}(t)} $$ Show that $F"(t)<0$.