inequality between matrix norms A is a $n\times k $ matrix.
I have to show that $\|A\|_2\leq \sqrt{\|A\|_1\cdot \|A\|_\infty}$. 
I know that $\|A\|_2^2 = \rho(A^H\cdot A)\leq \|A^H \cdot A\| $ for every $\| \cdot  \|$ submultiplicative matrix norm, but I don't know how to conclude.
Any idea? 

$\|A\|_{2}^2\leq trac(A^H\cdot A)\leq{\Vert A^H\cdot A\Vert_1\leq  \| A\|_1\|A\| _\infty}$. 
for more information see $D.46$ and $D.52$ of abstract harmonic analysis hewitt&ross pages 706 and 709
