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On the Bernstein-Szegö Theorem for Complex Polynomials

Published online by Cambridge University Press:  20 November 2018

D. A. Brannan  and
Ch. Pommerenke
D. A. Brannan
Affiliation:
University of Maryland, College Park, Maryland
Ch. Pommerenke
Affiliation:
Technische Universität, Berlin, Germany
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 370 - 371
Copyright
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