# When is Jensen's Inequality strict?

For a homework problem, I have to prove that for a random sample $$X_1, \ldots, X_n$$, drawn from a population with finite variance $$\sigma^2$$, with sample mean $$\bar{x}$$ and sample variance $$s^2$$, that $$E(S) \leq \sigma$$ and if $$\sigma^2 > 0$$, $$E(S) < \sigma$$.

The first part of the statement is a straightforward application of Jensen's Inequality: since $$g(x) = x^2$$ is convex everywhere, $$E(S^2) = \sigma^2 \geq (E(S))^2 \implies \sigma \geq E(S)$$.

However, I am not sure how to prove the second part of the statement. I got that since $$\sigma^2 > 0$$, then $$0 \geq E(S)$$, but I'm not sure if that helps. Are there any conditions that I just don't know about for Jensen's Inequality to be strict?