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Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

On support points of univalent functions and a disproof of a conjecture of Bombieri

Author(s): Richard Greiner; Oliver Roth
Journal: Proc. Amer. Math. Soc. 129 (2001), 3657-3664.
MSC (1991): Primary 30C70, 30C50; Secondary 30C35
Posted: May 3, 2001
MathSciNet review: 1860500
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Abstract | References | Similar articles | Additional information

Abstract:

We consider the linear functional $\operatorname{Re} (a_3+ \lambda a_2)$ for $\lambda \in i \mathbb{R}$ on the set of normalized univalent functions in the unit disk and use the result to disprove a conjecture of Bombieri.

References:

[Bom67]
E. Bombieri, On the local maximum property of the Koebe function, Inv. Math. 4 (1967), 26-67. MR 36:1635

[BH85]
D. Bshouty, W. Hengartner, Local behavior of coefficients in subclasses of $\mathcal{S}$, Contemp. Math. 38 (1985), 77-84. MR 86m:30007

[BH87]
D. Bshouty, W. Hengartner, A variation of the Koebe mapping in a dense subset of $\mathcal{S}$, Canad. J. Math. 39 (1987), no. 7, 54-73. MR 88j:30048.

[Bro81]
J. E. Brown, Univalent functions maximizing $\Real (a_3+\lambda a_2)$, Ill. J. Math. 25, No. 3 (1981), 446-454. MR 82j:30022

[Dur83]
P. L. Duren, Univalent Functions, Springer-Verlag, 1983. MR 85j:30034

[Haa83]
H. Haario, On the extreme points of classes of univalent functions, Ann. Acad. Sci. Fenn., Ser. A. I. Math. 8 (1983), 55-66. MR 84d:30019

[KT80]
R. A. Kortram, O. Tammi, Non-homogeneous combinations of coefficients of univalent functions, Ann. Acad. Sci. Fenn., Ser. A. I. Math. 5 (1980), 131-144. MR 81m:30017

[Leu85]
Y. I. Leung, Notes on Loewner differential equations, Contemp. Math. 38 (1985), 1-11.MR 86m:30007

[Pfl88]
A. Pfluger, On the functional $a_3-\lambda a_2^2$ in the class $\mathcal{S}$, Comp. Var. 10 (1988), 83-95. MR 89e:30034

[SSp50]
A. C. Schaeffer, D. C. Spencer, Coefficient Regions for Schlicht Functions, AMS Coll. Series, 1950. MR 12:326c

[Tam82]
O. Tammi, Extremum Problems for Bounded Univalent Functions, II, Springer LNM 913, 1982. MR 84a:30032

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Additional Information:

Richard Greiner
Affiliation: Department of Mathematics, Bayerische Julius-Maximilians-Universität, Am Hubland, D-97074 Würzburg, Germany
Email: greiner@mathematik.uni-wuerzburg.de

Oliver Roth
Affiliation: Department of Mathematics, Bayerische Julius-Maximilians-Universität, Am Hubland, D-97074 Würzburg, Germany
Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: roth@mathematik.uni-wuerzburg.de

DOI: 10.1090/S0002-9939-01-05994-9
PII: S 0002-9939(01)05994-9
Keywords: Univalent functions, support point, linear functional, fractional linear functional, Schiffer variation
Received by editor(s): December 28, 1999
Received by editor(s) in revised form: May 1, 2000
Posted: May 3, 2001
Additional Notes: This paper was completed while the second author was visiting the University of Michigan supported by a Feodor Lynen fellowship of the Alexander von Humboldt foundation. He thanks the faculty and staff for their hospitality.
Communicated by: Albert Baernstein II
Copyright of article: Copyright 2001, American Mathematical Society




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