Prove $\int_Rfg\,dm\leq\|f\|_p^{1-p/r}\|g\|_p^{1-q/r}(\int_Rf^pg^q\,dm)^{1/r}$, where $1\leq p\leq\infty$ and $\frac1{r}=\frac1{p}+\frac1{q}-1$

Question

Let $f$, $g$ be positive real functions. And $f \in L^p(R)$, $g \in L^q(R)$, and $1 \leqslant p,q <\infty$. Then $fg \in L^1(R)$ and $$ \int_R fg \,dm\;\leqslant\; \|f\|_p^{1-p/r}\|g\|_p^{1-q/r}\left(\int_R f^pg^q \,dm\right)^{1/r}$$ Where $$1\leqslant p \leqslant +\infty \quad\text{and}\quad\frac{1}{r}=\frac{1}{p}+\frac{1}{q}-1$$

Answer 1

Hints: I suppose $1+\frac 1 r =\frac 1 p +\frac 1 q$. Write $fg$ as $f^{p/r}g^{q/r} f^{1-p/r}g^{1-q/r}$ and apply Holder with conjugate indices $r$ and $\frac r {r-1}$. You will get $(\int f^{p}g^{q})^{1/r}$ as one of the factors. Now applying another Holder to complete the proof is fairly straightfoward.