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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?

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

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  • $\begingroup$ Me too i don't know but if we repeat the same steps in the previous proof as i said, we find the result, For LHS is positive so does the RHS and it depends on x what's the problem o that? $\endgroup$ – Cicilio May 8 '16 at 18:51

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