Inequality between norm 1,norm 2 and norm $\infty$ of Matrices

Suppose $A$ is a $m\times n$ matrix.

Then Prove that, $\begin{equation*} \|A\|_2\leq \sqrt{\|A\|_1 \|A\|_{\infty}} \end{equation*}$

I have proved the following relations: $\begin{align*} \frac{1}{\sqrt{n}}\|A\|_{\infty}\leq \|A\|_2\leq\sqrt{m}\|A\|_{\infty}\\ \frac{1}{\sqrt{m}}\|A\|_{1}\leq \|A\|_2\leq\sqrt{n}\|A\|_{1} \end{align*}$ Also I feel that somehow Holder's inequality for the special case when $p=1$ and $q=\infty$ might be useful.But I couldn't prove that.

Edit: I would like to have a prove that do not use the information that $\|A\|_2=\sqrt{\rho(A^TA)}$

Usage of inequalities like Cauchy Schwartz or Holder is fine.