The Schur norm of a matrix $A$ is defined to be $\|A\|_S=\max\{\|A\circ X\|: \|X\|\leq 1\}$, where $\|\cdot \|$ is the operator norm of a matrix, i.e., the largest singular value.
Let $a_1,\ldots, a_m, b_1,\ldots, b_n$ be positive reals.Let $A$ be an $m\times n$ matrix defined to be $A_{i,j}=(a_i-b_j)/(a_i+b_j)$.
My question is how to compute $\|A\|_S$. Is it upper bounded by an absolute constant independent of $m, n$?