MATH. SCAND. 104 (2009), 147–160 INEQUALITIES FOR PRODUCTS OF POLYNOMIALS I

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. 1. The problem and its history Let E be a compact set in the complex plane C. For a function f: E → C define the uniform (sup) norm as follows: ‖f ‖E = sup z∈E |f (z)|. Clearly ‖f1f2‖E ≤ ‖f1‖E ‖f2‖E, but this inequality is not reversible, in gen-eral, not even with a constant factor in front of the right hand side. Indeed, ‖f1‖E ‖f2‖E ≤ C ‖f1f2‖E does not hold for functions with disjoint sup-ports in E, for example. However, the situation is quite different for algebraic polynomials {pk(z)}mk=1 and their product p(z):= ∏m k=1 pk(z). Polynomial inequalities of the form (1.1) m∏ k=1 ‖pk‖E ≤ C‖p‖E, exist and are readily available. One of the first results in this direction is due to Kneser [19], for E = [−1, 1] and m = 2 (see also Aumann [1]), who proved that (1.2) ‖p1‖[−1,1]‖p2‖[−1,1] ≤ K,n‖p1p2‖[−1,1], deg p1 = , deg p2 = n −