Arkiv för Matematik

  • Ark. Mat.
  • Volume 36, Number 2 (1998), 255-273.

Coefficient estimates for negative powers of the derivative of univalent functions

Daniel Bertilsson

Full-text: Open access

Abstract

Letf be a one-to-one analytic function in the unit disc with f′(0)=1. We prove sharp estimates for certain Taylor coefficients of the functions (f′)p, where p<0. The proof is similar to de Branges’ proof of Bieberbach’s conjecture, and thus relies on Löwner’s equation. A special case leads to a generalization of the usual estimate for the Schwarzian derivative of f. We use this to improve known estimates for integral means of the functions |f′|p for integers p⪯−2.

Article information

Source
Ark. Mat., Volume 36, Number 2 (1998), 255-273.

Dates
Received: 4 August 1997
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898602

Digital Object Identifier
doi:10.1007/BF02384769

Mathematical Reviews number (MathSciNet)
MR1650434

Zentralblatt MATH identifier
1025.30013

Rights
1998 © Institut Mittag-Leffler

Citation

Bertilsson, Daniel. Coefficient estimates for negative powers of the derivative of univalent functions. Ark. Mat. 36 (1998), no. 2, 255--273. doi:10.1007/BF02384769. https://projecteuclid.org/euclid.afm/1485898602


Export citation

References

  • Ahlfors, L. V., Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill, New York, 1973.
  • Bol, G., Invarianten linearer Differentialgleichungen, Abh. Math. Sem. Univ. Hamburg 16 (1949), 1–28.
  • Brennan, J. E., The integrability of the derivative in conformal mapping, J. London Math. Soc. 8 (1978), 261–272.
  • Carleson, L. and Makarov, N. G., Some results connected with Brennan’s conjecture, Ark. Mat. 32 (1994), 33–62.
  • de Branges, L., A proof of the Bieberbach conjecture, Acta Math. 154 (1985), 137–152.
  • Duren, P. L., Univalent Functions, Springer-Verlag, New York, 1983.
  • Fitzgerald, C. H. and Pommerenke, C., The de Branges theorem on univalent functions, Trans. Amer. Math. Soc. 290 (1985), 683–690.
  • Gnuschke-Hauschild, D. and Pommerenke, C., On Bloch functions and gap series, J. Reine Angew. Math. 367 (1986), 172–186.
  • Gustafsson, B. and Peetre, J., Möbius invariant operators on Riemann surfaces, in Function Spaces, Differential Operators and Nonlinear Analysis (Sodankylä, 1988) (Päivärinta, L., ed.), Pitman Res. Notes Math. Ser. 211, pp. 14–75, Longman, Harlow, 1989.
  • Gustafsson, B. and Peetre, J., Notes on projective structures on complex manifolds, Nagoya Math. J. 116 (1989), 63–88.
  • Ozawa, M., On certain coefficient inequalities of univalent functions, Ködai Math. Sem. Rep. 16 (1964), 183–188.
  • Pommerenke, C., Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.
  • Pommerenke, C., Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin-Heidelberg, 1992.