Upper bound on E[ln(X)^2]

Question

The question is to state, for positive $X$, when $\mathbb{E}[(\ln X)^2] \leq (\ln\mathbb{E}X)^2$.

I think this is asking to use Jensen's inequality, but this inequality says the expected value is the upper bound: for integrable $X$ and convex $g: \mathbb{R}\rightarrow\mathbb{R}$, $g(\mathbb{E}X) \leq \mathbb{E}g(X)$.

Something is wrong with my reasoning: $y^2$ is convex, so why isn't $(\ln\mathbb{E}X)^2 \leq \mathbb{E}[(\ln X)^2]$ (for $X > 0$)?