Let $p\in(1,\infty)$ and $(X,\mathcal{F},\mu)$ a measure space such that $\mu(X)\not=0$. Let $f,g:X\to\mathbb{R}$ be such that $g\not=0$ a.e., $\|fg\|_1<\infty$ and $\|g\|_{-\frac{1}{p-1}}<\infty$. Then prove:$$\|fg\|_1\geq\| f\|_{\frac{1}{p}}\|g\|_{-\frac{1}{p-1}}$$ I tried searching for "reverse holder inequality" and I found I thousand things that do not look similar to this. I just wrote the definitions of the norms and couldn't do nothing because the only thing I thought that I could use was Holder inequality but that is in the reverse direction of the inequality above (laugh). I would aprecciate some hint to how I should proceed. Login To add answer/comment