Suppose $X>0$ is a random variable with $\mathbb{E}X=c$ and $\mathbb{E}X^{-1}=c^{-1}$.
By applying Cauchy-Schwarz to $\sqrt{X}$ and $\sqrt{X^{-1}}$ and using the "equality iff linearly dependent" case, we have that $X$ must be constant.
But this seems like far too heavy machinery for what ought to be an obvious statement. Is there a simpler proof? Is there an "equality iff" form of Jensen's inequality?