Extension of Bernšteĭn's theorem

Author:
S. H. Tung

Journal:
Proc. Amer. Math. Soc. **83** (1981), 103-106

MSC:
Primary 32A15; Secondary 30E10, 32A30

DOI:
https://doi.org/10.1090/S0002-9939-1981-0619992-9

MathSciNet review:
619992

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Abstract | References | Similar Articles | Additional Information

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

**[1]**L. V. Ahlfors,*Complex analysis*, 2nd ed., McGraw-Hill, New York, 1966. MR**510197 (80c:30001)****[2]**E. Hille,*Analytic function theory*, Vol. II, Ginn, Boston, Mass., 1962. MR**0201608 (34:1490)****[3]**L. Hörmander,*On a theorem of Grace*, Math. Scand.**2**(1954), 55-64. MR**0062844 (16:27b)****[4]**O. D. Kellogg,*On bounded polynomials in several variables*, Math. Z.**27**(1928), 55-64. MR**1544896****[5]**M. A. Malik,*On the derivative of a polynomial*, J. London Math. Soc. (2)**1**(1969), 57-60. MR**0249583 (40:2827)****[6]**M. Marden,*Geometry of polynomials*, Math. Surveys, no. 3, Amer. Math. Soc., Providence, R. I., 1966. MR**0225972 (37:1562)**

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Additional Information

DOI:
https://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