One form of Jensen's inequality for the finite case, tells us that
$$ \sum_{x \in X} p(x) \log q(x) \leq \log\sum_{x \in X} p(x) \cdot q(x) $$
For positive p(x), and $\sum_{x \in X} p(x) = 1$, $q(x)$ real, and $X$ finite. I am using the $\log$, but any concave function could be substituted.
or the probabilistic version:
$$ \mathbb{E}( \log X) \leq \log \mathbb{E}(X)$$ Where $\mathbb{E}$ is the expectation of $X$.
However, is this inequality true for countable $X$? The book I'm reading (elements of information theory, 2006), seems to prove it for the finite case, but uses the countable case without mentioning it.
Also on wikipedia the it seems the first inequality in my post is only for the finite case, whereas the probablistic verson makes no mention of cardinality of the probability space.
Sketch, for the case where $X$ has support over $\mathbb{N}$:
Consider the variables $Y_n= \min(X,n)$ for $n=1,2 \cdots$
Then $Y_n$ has finite support, and so we have $\mathbb{E}( \log Y_n) \leq \log \mathbb{E} (Y_n)$
But $Y_n$ converges in distribution to $X$ as $n\to \infty$. Then $\log \mathbb{E}( Y_n) \to \log \mathbb{E}( X) $ and $\mathbb{E}( \log Y_n)\to \mathbb{E}( \log X) $. This implies $\mathbb{E}( \log X) \leq \log \mathbb{E} (X)$