Extension of Bernšteĭn's theorem
Author:
S. H. Tung
Journal:
Proc. Amer. Math. Soc. 83 (1981), 103106
MSC:
Primary 32A15; Secondary 30E10, 32A30
MathSciNet review:
619992
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A wellknown 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]
Lars
V. Ahlfors, Complex analysis, 3rd ed., McGrawHill 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
(80c:30001)
 [2]
Einar
Hille, Analytic function theory. Vol. II, Introductions to
Higher Mathematics, Ginn and Co., Boston, Mass.New YorkToronto, Ont.,
1962. MR
0201608 (34 #1490)
 [3]
Lars
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), no. 1, 55–64. MR
1544896, http://dx.doi.org/10.1007/BF01171085
 [5]
M.
A. Malik, On the derivative of a polynomial, J. London Math.
Soc. (2) 1 (1969), 57–60. MR 0249583
(40 #2827)
 [6]
Morris
Marden, Geometry of polynomials, Second edition. Mathematical
Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972
(37 #1562)
 [1]
 L. V. Ahlfors, Complex analysis, 2nd ed., McGrawHill, 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), 5564. MR 0062844 (16:27b)
 [4]
 O. D. Kellogg, On bounded polynomials in several variables, Math. Z. 27 (1928), 5564. MR 1544896
 [5]
 M. A. Malik, On the derivative of a polynomial, J. London Math. Soc. (2) 1 (1969), 5760. 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)
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/S00029939198106199929
PII:
S 00029939(1981)06199929
Keywords:
Bernstein's theorem,
polynomial,
the unit ball in several complex variables
Article copyright:
© Copyright 1981
American Mathematical Society
