Given an undirected graph $G = (V,E)$, where $|V| = n$ and $|E| = m$, I am trying to show that $\sum_{v \in V} \deg(v)^2 \in O(m^2/n)$.
Using Jensen's inequality, we can show that $$\sum_{v \in V} \deg(v)^2 \geq n \left(\frac{1}{n} \sum_{v \in V} \deg(v) \right)^2 = 4m^2/n \in \Omega(m^2/n),$$ however, since $f(x) = x^2$ is convex (and not concave), I can't use Jensen's inequality to show the upper bound.
It's not true. Take a star graph on $n$ vertices, which has $m = n - 1$ edges, which makes $m^2/n = \Theta(n)$. However:
$$\sum \text{deg}(v)^2 = 1^2 + 1^2 + \dots + 1^2 + (n-1)^2 = n(n-1) = \Theta(n^2)$$