Suppose $m(E)=1$ and $f,g \geq 0$ with $fg \geq 1$. Show that $\displaystyle \int_E f dm \int_E g dm \geq 1$
Can someone please point me to the right direction?
Suppose $m(E)=1$ and $f,g \geq 0$ with $fg \geq 1$. Show that $\displaystyle \int_E f dm \int_E g dm \geq 1$
Can someone please point me to the right direction?
Hint : Use Cauchy-Scharwz inequality.
Hint 2 : If $f,g \geq 0, fg \geq 1$ then $\sqrt{f}\sqrt{g} \geq 1$ also.
I assume that $f$ and $g$ are integrable. Outside a null-set we have $g \geq \frac{1}{f} \gt 0$, in particular $\frac{1}{f}$ is integrable. Now apply Jensen's inequality $\varphi\left(\int f\right) \leq \int \varphi \circ f$ for the convex function $\varphi(x) = \frac{1}{x}$ (here we use $f \geq 0$ again), so $\frac{1}{\int f} \leq \int \frac{1}{f} \leq \int g$.