An optimization problem concerning multiplicative functions

In this paper we study the problem of maximizing a quadratic form 〈Ax,x〉〈Ax,x〉 subject to ‖x‖q=1‖x‖q=1, where A   has matrix entries f([i,j](i,j)) with i,j|ki,j|k and q≥1q≥1. We investigate when the optimum is achieved at a ‘multiplicative’ point; i.e. where x1xmn=xmxnx1xmn=xmxn. This turns out to depend on both f and q, with a marked difference appearing as q varies between 1 and 2. We prove some partial results and conjecture that for f   multiplicative such that 0