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On the Bernstein-Szegö Theorem for Complex Polynomials
Published online by Cambridge University Press: 20 November 2018
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Let p(z) be a complex polynomial of degree less than or equal to n. Generalizing the well-known Bernstein theorem, Szegö (3) has shown that
We shall give a partial generalization of this result.
THEOREM. Let p(z) be a polynomial of degree at most n. Let R be the radius of the largest disc contained in G = {p(z): |z| < 1}. Then
Since R ≦ max|2|=1 |Re p(z)|, we obtain Szegö's result, but with a worse constant. It would be interesting to see whether it is possible to replace the constant e by 1. If so, Rzn would be an extremal for all n, and 
another extremal for even n.
The proof is based on the following result of Ahlfors (1) (cf., e.g., 2, p. 321). This result corresponds to the estimate f or the Landau constant.
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- Research Article
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- Copyright © Canadian Mathematical Society 1969
References
1.
Ahlfors, L., An extension of Schwarz's lemma, Trans. Amer. Math. Soc.
43 (1938), 359–364.Google Scholar
2.
Golusin, G. M., Geometrische Funktionentheorie, Hochschulbiïcher fur Mathematik, Bd. 31 (VEB Deutscher Verlag der Wissenschaften, Berlin, 1957).Google Scholar
3.
Szego, G., Über einen Satz des Herrn Serge Bernstein, Schr. Kônigsb. gelehrt. Ges. Natur. Kl.
5 (1928), 59–70.Google Scholar
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