  We study inequalities connecting the product of uniform norms of polynomials
with the norm of their product. This circle of problems include the
Gelfond-Mahler inequality for the unit disk and the Kneser-Borwein inequality
for the segment $[-1,1]$. Furthermore, the asymptotically sharp constants are
known for such inequalities over arbitrary compact sets in the complex plane.
It is shown here that this best constant is smallest (namely: 2) for a disk. We
also conjecture that it takes its largest value for a segment, among all
compact connected sets in the plane.
