# Proving Hölder's Inequality with variable exponent

My question is it correct this inequality ? $$\int_\Omega \vert f(x)g(x) \vert \, dx \leq \left( \int_\Omega \vert f(x)\vert^{p(x)} \, dx\right)^{\frac{1}{p(x)}} \left( \int_\Omega \vert g(x)\vert^{q(x)} \, dx \right)^{\frac{1}{q(x)}},\quad \frac{1}{p(x)}+\frac{1}{q(x)}=1.$$ My idea is that we use the idea of the proof in the previons question "Proving Hölder's Inequality" and with the same proof, we get it?

The inequality should not be true in general since the LHS is independent of $x$ while the RHS does depend on $x$.