It certainly holds for all matrices where $|a_{ij}|=|a_{ji}|$ (meaning both symmetric and skew-symmetric matruces, among otbers). It is reported here that

$||A||_2^2\le||A||_1||A||_\infty,$

and where the condition above is satisfied $||A||_1=||A||_\infty$.

It certainly holds for all matrices where $|a_{ij}|=|a_{ji}|$ (meaning both symmetric and skew-symmetric matruces, among otbers). It is reported here that

$||A||_2^2\le||A||_1||A||_\infty,$

and where the condition above is satisfied $||A||_1=||A||_\infty$.

For all $n×n$ matrices we always have

$||A||_1\le n||A||_\infty$

$||A||_\infty\le n||A||_1$

and so the bound given above implies

$||A||_2\le\sqrt{n}||A||_1$

$||A||_2\le\sqrt{n}||A||_\infty.$