Hölders inequality is $\int |fg|dx \le||f||_p||g||_q$
Define $F(x)= \frac{f(x)}{(g(x))^{q/p}}$ and $\nu dx =g(x)^q$
and $\Phi(t)= |t|^p$, $p\in (1,\infty)$
Now lets find $\Phi(\int F(x) d\nu)= \frac{(\int fg)^p}{||g||_q^{qp}}$ and similarly find $\int \Phi (F(x))$ and apply jensens inequality .
My doubt is in the setting of $F(x)$ and $\nu dx $ Can anyone help me to make this proof correct .
Thanks