Sharp Inequalities for the Product of Polynomials

Let f 1 ( z ),…, f m ( z ) be polynomials with complex coefficients, and let their product be of degree n . For any polynomial, let ‖ f ‖ be the maximum of | f ( z )| on the unit circle. We determine constants C m < 2 for which‖f 1 ‖· · · ‖f m ‖⩽ C m n ‖f 1 · · · f m ‖for any n . The inequalities are asymptotically sharp as n →∞. This improves earlier results of Gel'fond and Mahler, who gave the constants e and 2 respectively. If f 1 ,…, f m have real coefficients, we show that‖f 1 ‖· · · ‖f m ‖⩽ C 2 n ‖f 1 … f m ‖for all m ⩾ 2 and that this is asymptotically sharp. That is, in the real case, the best constant does not depend upon m for m ⩾ 2.