Convexity of $A \mapsto |A^{T}A|^{2}$

Question

This is an extension to Convexity of $|A^TA|$. Let $A \in \mathbb{R}^{m \times n}$ and let $|\cdot|$ denote the Frobenius (matrix) norm. Define function $f : \mathbb{R}^{m \times n} \to \mathbb{R}$ as $f(A) := |A^{T} A|^{2}$. Is $f$ convex?

Answer 1

$f(x)=x^2$ is convex and increasing, also $g(A) = \lvert A^T A\rvert$ is convex, therefore $h(A)=f(g(A)) = \lvert A^T A\rvert^2$ is convex by composition rules.