Extension of Bernšteĭn’s theorem

Tung, S. H.

doi:10.1090/S0002-9939-1981-0619992-9

Extension of Bernšteĭn’s theorem
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by S. H. Tung PDF

Proc. Amer. Math. Soc. 83 (1981), 103-106 Request permission

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

Lars V. Ahlfors, Complex analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable. MR 510197
Einar Hille, Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Company, Boston, Mass.-New York-Toronto, Ont., 1962. MR 0201608
Lars Hörmander, On a theorem of Grace, Math. Scand. 2 (1954), 55–64. MR 62844, DOI 10.7146/math.scand.a-10395
O. D. Kellogg, On bounded polynomials in several variables, Math. Z. 27 (1928), no. 1, 55–64. MR 1544896, DOI 10.1007/BF01171085
M. A. Malik, On the derivative of a polynomial, J. London Math. Soc. (2) 1 (1969), 57–60. MR 249583, DOI 10.1112/jlms/s2-1.1.57
Morris Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972