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On an Inequality of Bombieri

Published online by Cambridge University Press:  09 April 2009

D. C. Peaslee  and
W. A. Coppel
D. C. Peaslee
Affiliation:
Department of Theoretical Physics and Mathematics Institue of Advanced Studies The Australian National UniversityCanberra, A.C.T.
W. A. Coppel
Affiliation:
Department of Theoretical Physics and Mathematics Institue of Advanced Studies The Australian National UniversityCanberra, A.C.T.
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Bieberbach's conjecture, proposed in 1916 and still unsolved, states that if ƒ(z) = z+a2z2+… is holomorphic and univalent in the disc ∣z∣ < 1 then ∣an∣ ≦ n for each n ≧ 2, with equality for some n only if ƒ(z) is the Koebe function of is obtained from this function by a rotation. Very recently Bombieri has succeeded in showing that if ƒ(z) is sufficiently close to the Koebe function, then with equality only if ƒ(z) = k(z). This had previously been proved by Garabedian, Ross and Schiffer [3] for even values of n.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 9 , Issue 3-4 , May 1969 , pp. 399 - 404
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Bombieri, E., ‘Sulla seconda variazione della funzione di Koebe’, Boll. Un. Mat. Ital. 22 (1967), 25–32.Google Scholar
[2]Erdélyi, A., et al. , Higher transcendental functions (McGraw-Hill, New York, 1953), Vol. 2, p. 223.Google Scholar
[3]Garabedian, P. R., Ross, G. G. and Schiffer, M., ‘On the Bieberbach conjecture for even n’, J. Math. Mech. 14 (1965), 975–989.Google Scholar