Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-27T02:05:10.216Z Has data issue: false hasContentIssue false
  • English
  • Français

On Generalized Schwarz-Pick Estimates

Part of: General properties Special classes of linear operators

Published online by Cambridge University Press:  21 December 2009

J. M. Anderson  and
J. Rovnyak
J. M. Anderson
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, U.K.
J. Rovnyak
Affiliation:
Department of Mathematics, University of Virginia, P.O. Box 400137, Charlottesville, VA 22904–4137, U.S.A. E-mail: [email protected]
Get access

Extract

By Pick's invariant form of Schwarz's lemma, an analytic function B (z) which is bounded by one in the unit disk D = {z: |z| < 1} satisfies the inequality

at each point α of D. Recently, several authors [2, 10, 11] have obtained more general estimates for higher order derivatives. Best possible estimates are due to Ruscheweyh [12]. Below in §2 we use a Hilbert space method to derive Ruscheweyh's results. The operator method applies equally well to operator-valued functions, and this generalization is outlined in §3.

MSC classification

Secondary: 47B32: Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) 30A10: Inequalities in the complex domain
Type
Research Article
Information
Mathematika , Volume 53 , Issue 1 , June 2006 , pp. 161 - 168
Copyright
Copyright © University College London 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Alpay, D., Dijksma, A., Rovnyak, J., and de Snoo, H., Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Operator Theory: Advances and Applications 96, Birkhäuser Verlag (Basel, 1997).Google Scholar
2Bénéteau, C., Dahlner, A., and Khavinson, D., Remarks on the Bohr phenomenon, Comput. Methods Funct. Theory 4 (2004), no. 1, 1–19.CrossRefGoogle Scholar
3Bénéteau, C. and Korenblum, B., Some coefficient estimates for Hp functions, Complex Analysis and Dynamical Systems, Contemp. Math. 364, Amer. Math. Soc. (Providence, RI, 2004), 5–14.CrossRefGoogle Scholar
4Bohr, H., A theorem concerning power series, Proc. London Math. Soc. (2) 13 (1914), 1–5. (In Collected Mathematical Works Vol. III, paper #E 3, Dansk Matematisk Forening (København, 1952.))Google Scholar
5de Branges, L. and Rovnyak, J., Canonical models in quantum scattering theory, Perturbation Theory and its Applications in Quantum Mechanics (Proc. Adv. Sem. Math. Res. Center, U.S. Army, Theoret. Chem. Inst., Univ. of Wisconsin, Madison, Wis., 1965), Wiley (New York, 1966), 295–392.Google Scholar
6de Branges, L. and Rovnyak, J., Square Summable Power Series, Holt, Rinehart and Winston (New York, 1966).Google Scholar
7Khavinson, S. Ya., Two papers on extremal problems in complex analysis, Amer. Math. Soc. Transl. (2), 129, American Mathematical Society, (Providence, RI, 1986). (Translated from the Russian manuscript by D. Khavinson.)Google Scholar
8Kreĭn, M. G. and Langer, H., Über die verallgemeinerten Resolventen und die charakteristische Funktion eines isometrischen Operators im Raume Πκ, Hilbert Space Operators and Operator Algebras (Proc. Internat. Conf., Tihany, 1970), North-Holland, (Amsterdam, 1972), 353–399. (Colloq. Math. Soc. János Bolyai 5.)Google Scholar
9Landau, E. and Gaier, D., Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie (3rd ed.), Springer-Verlag (Berlin, 1986).CrossRefGoogle Scholar
10MacCluer, B. D., Stroethoff, K., and Zhao, R., Generalized Schwarz-Pick estimates, Proc. Amer. Math. Soc. 131 (2003), 593–599 (electronic).CrossRefGoogle Scholar
11MacCluer, B. D., Schwarz-Pick type estimates, Complex Var. Theory Appl. 48 (2003), 711–730.Google Scholar
12Ruscheweyh, St., Two remarks on bounded analytic functions, Serdica 11 (1985), 200–202.Google Scholar
13Sarason, D., Sub-Hardy Hilbert Spaces in the Unit Disk, University of Arkansas Lecture Notes in the Mathematical Sciences 10, John Wiley & Sons Inc. (New York, 1994).Google Scholar