It seems worth mentioning that the inequality  $$\langle r\rangle ~\langle\frac{1}{r}\rangle ~>~1$$ follows directly from the Cauchy-Schwarz inequality $$ || \sqrt{r}\psi ||~ || \frac{1}{\sqrt{r}}\psi ||~>~ || \psi ||^2~=~1$$ when the two functions $\sqrt{r}\psi $ and $\frac{1}{\sqrt{r}}\psi$ are not proportional.

It seems worth mentioning that the inequality 

$$\langle r\rangle ~\langle\frac{1}{r}\rangle ~>~1$$

follows directly from the Cauchy-Schwarz inequality

$$|| \sqrt{r}\psi ||~ || \frac{1}{\sqrt{r}}\psi ||~>~ || \psi ||^2~=~1$$

when the two functions $\sqrt{r}\psi $ and $\frac{1}{\sqrt{r}}\psi$ are not proportional. Here we have used the fact that an expectation value

$$\langle f\rangle~=~\int_{\mathbb{R}^3}\! d^3r~f|\psi|^2=||\sqrt{f} \psi ||^2$$

of a non-negative function $f$ is related to the 2-norm $$ || \psi ||~:=~\sqrt{\int_{\mathbb{R}^3}\! d^3r~|\psi|^2} . $$

It seems worth mentioning that the inequality $$\langle r\rangle \langle\frac{1}{r}\rangle ~>~1$$ follows directly from Cauchy-Schwarz inequality  $$ || \sqrt{r}\psi ||~ || \frac{1}{\sqrt{r}}\psi ||~>~ || \psi ||^2~=~1$$ when $\sqrt{r}\psi $ and $\frac{1}{\sqrt{r}}\psi$ are not proportional.

It seems worth mentioning that the inequality $$\langle r\rangle ~\langle\frac{1}{r}\rangle ~>~1$$ follows directly from the Cauchy-Schwarz inequality  $$ || \sqrt{r}\psi ||~ || \frac{1}{\sqrt{r}}\psi ||~>~ || \psi ||^2~=~1$$ when the two functions $\sqrt{r}\psi $ and $\frac{1}{\sqrt{r}}\psi$ are not proportional.