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Bulletin of the London Mathematical Society 1994 26(5):449-454; doi:10.1112/blms/26.5.449
© 1994 by London Mathematical Society
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© London Mathematical Society

Sharp Inequalities for the Product of Polynomials

David W. Boyd

Department of Mathematics University of British Columbia Vancouver Canada V6T 1Z2

Received 29 January 1992. Revision received 28 June 1993.

Let f1(z),..., fm(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 Cm < 2 for which Formula for any n. The inequalities are asymptotically sharp as n ->{infty}. This improves earlier results of Gel'fond and Mahler, who gave the constants e and 2 respectively. If f1,..., fm have real coefficients, we show that Formula 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.


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