## does this integral inequality of a convex function hold?

let $\phi$ be a convex function with $\phi(0)=0$, $\phi^{-1}$ be the inverse function of $\phi$, ${\phi^{-1}}^{'}$ denotes the derivation of $\phi^{-1}$, then does the follow inequality hold? $\int_{S^{n-1}}{\phi^{-1}}^{'}(\phi(u))\phi(u)dS(u)\geq\int_{S^{n-1}}{\phi^{-1}}^{'}(\phi(u))dS(u)\int_{S^{n-1}}\phi(u)dS(u)$?

for example: for $\phi(x)=x^p$ then the above inequality is just the holder inequality, up some constant.

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 Hah? Is $\phi$ a real valued function of a vector argument? If so, what is $\phi^{-1}$? If not, how do you compute $\phi(u)$? – fedja Jun 2 at 1:10