I would like to know if there is any relationship between Fatou's lemma and Jensen's inequality. On paper, I find a similarity in their expressions.
Could someone provide me with some insight explaining this similarity?
The concave function $f$ is an pointwise infimum of a countable family $(a_n)$ of affine functions. You can arrange for the sequence $(f_n)$ to contain each $a_n\circ X$ infinitely often, so that $\liminf_{n\to\infty} f_n(\omega)=\inf_n a_n(X(\omega))$. Now the left hand sides of the two inequalities are the same, and so are the right hand sides, since the affinity of $a_n$ implies $E(f_n)=E(a_n\circ X)=a_n(E(X))$.
However, Fatou's lemma contains much more information, as there is really no limit operation in this construction, just the infimum. So the connection is pretty much one way.
Here is an alternative way to think about it, inspired by Harald's answer. Consider the following lemma: for a family of functions $f_{n}$ and a random variable $X$, $$ \mathbb{E}[\inf_{n} f_{n}(X)] \leq \inf_{n} \mathbb{E}[f_{n}(X)]. $$
This is true because $ \mathbb{E}[\inf_{n} f_{n}(X)] \leq \mathbb{E}[f_{k}(X)]$ for each $k$.
Now, Jensen's inequality is a direct consequence of this, because you can represent any concave function as an infimum of affine functions (in fact, this characterizes concave functions). That combined with the linearity of expectation/integration gives the desired result.
Similarly, Fatou's lemma follows from this and the monotone convergence theorem. In particular, note that \begin{align*} \lim_{n} \inf_{k \geq n} \mathbb{E}[f_{n}(X)] &\geq \lim_{n} \mathbb{E}[ \inf_{k \geq n} f_{k}(X)] \\ &= \mathbb{E}[ \lim_{n} \inf_{k \geq n} f_{k}(X)], \end{align*}
where the equality follows from the fact that $(\inf_{k \geq n} f_{k} : n)$ is a monotonely increasing sequence of functions.
It is interesting to notice from the proof that the inequality in Fatou's lemma comes entirely from the nature of the infimum and not the limit part.
