Lower bound for expectation of squared log?

Question

Is there a (tight) lower bound for $\mathbb{E}[(\log x)^2]$ where $x$ is a non-negative random variable? Jensen's inequality doesn't seem to apply here since the squared of a log isn't convex. Thanks!

Answer 1

Imagine $X$ takes value $a>0$ w.p. $p_a$ and $\frac{1}{a}$ w.p. $1-p_a$

Then, $\log(X)^2 = \log(a)^2=log(\frac{1}{a})^2$; hence, $E[\log(X)^2]$ is a constant whose value is $\log(a)^2$.

Now, lets look at what we can say just about this special case:

$\lim\limits_{a \to \infty} \log(a)^2 = \infty$ and $\lim\limits_{a \to 0} \log(a)^2= \infty$

so even in this special case, the expected value of the squared log of an arbitrary positive rv has no finite bound.