Show that $ \lVert A \rVert_2^2 \leq \lVert A \rVert _1 \lVert A \rVert _ \infty $
Let $\|\cdot\|$ be any matrix norm induced by a vector norm. Then we have $$\|A\|_2^2= \rho(AA^H) \leq \|AA^H\| \leq \|A\|\|A^H\|.$$ Here the first inequality follows from a "famous theorem" (see e.g. Proposition 4.4) and the second inequality follows from the fact that $\|\cdot\|_\infty$ is a matrix norm induced by a vector norm and thus is submultiplicative. Finally note that $\|A\|_1 =\|A^H\|_\infty$.