Geodesic
On the Bernstein-Szegö Theorem for Complex Polynomials
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 370-371

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Brannan, D. A.; Pommerenke, Ch. On the Bernstein-Szegö Theorem for Complex Polynomials. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 370-371. doi: 10.4153/CJM-1969-037-4
@article{10_4153_CJM_1969_037_4,
     author = {Brannan, D. A. and Pommerenke, Ch.},
     title = {On the {Bernstein-Szeg\"o} {Theorem} for {Complex} {Polynomials}},
     journal = {Canadian journal of mathematics},
     pages = {370--371},
     year = {1969},
     volume = {21},
     number = {1},
     doi = {10.4153/CJM-1969-037-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1969-037-4/}
}
TY  - JOUR
AU  - Brannan, D. A.
AU  - Pommerenke, Ch.
TI  - On the Bernstein-Szegö Theorem for Complex Polynomials
JO  - Canadian journal of mathematics
PY  - 1969
SP  - 370
EP  - 371
VL  - 21
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1969-037-4/
DO  - 10.4153/CJM-1969-037-4
ID  - 10_4153_CJM_1969_037_4
ER  - 
%0 Journal Article
%A Brannan, D. A.
%A Pommerenke, Ch.
%T On the Bernstein-Szegö Theorem for Complex Polynomials
%J Canadian journal of mathematics
%D 1969
%P 370-371
%V 21
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1969-037-4/
%R 10.4153/CJM-1969-037-4
%F 10_4153_CJM_1969_037_4

[1] 1. Ahlfors, L., An extension of Schwarz's lemma, Trans. Amer. Math. Soc. 43 (1938), 359–364. Google Scholar

[2] 2. Golusin, G. M., Geometrische Funktionentheorie, Hochschulbiïcher fur Mathematik, Bd. 31 (VEB Deutscher Verlag der Wissenschaften, Berlin, 1957). Google Scholar

[3] 3. Szego, G., Über einen Satz des Herrn Serge Bernstein, Schr. Kônigsb. gelehrt. Ges. Natur. Kl. 5 (1928), 59–70. Google Scholar

Cité par Sources :