Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Extension of Bernšteĭn's theorem


Author: S. H. Tung
Journal: Proc. Amer. Math. Soc. 83 (1981), 103-106
MSC: Primary 32A15; Secondary 30E10, 32A30
MathSciNet review: 619992
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A well-known theorem of Bernstein states that if a polynomial of degree $ n$ of a complex variable has its modulus no larger than one on the unit disk then the modulus of its derivative will not exceed $ n$ on the unit disk. Here we extend the theorem to polynomials on the unit ball in several complex variables.


References [Enhancements On Off] (What's this?)

  • [1] Lars V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. MR 510197
  • [2] Einar Hille, Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass.-New York-Toronto, Ont., 1962. MR 0201608
  • [3] Lars Hörmander, On a theorem of Grace, Math. Scand. 2 (1954), 55–64. MR 0062844
  • [4] O. D. Kellogg, On bounded polynomials in several variables, Math. Z. 27 (1928), no. 1, 55–64. MR 1544896, 10.1007/BF01171085
  • [5] M. A. Malik, On the derivative of a polynomial, J. London Math. Soc. (2) 1 (1969), 57–60. MR 0249583
  • [6] Morris Marden, Geometry of polynomials, Second edition. Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 32A15, 30E10, 32A30

Retrieve articles in all journals with MSC: 32A15, 30E10, 32A30


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0619992-9
Keywords: Bernstein's theorem, polynomial, the unit ball in several complex variables
Article copyright: © Copyright 1981 American Mathematical Society