Computational Methods and Function Theory

, Volume 12, Issue 2, pp 483–499 | Cite as

On a Multi-Point Schwarz-Pick Lemma



We consider the multi-point Schwarz-Pick Lemma and its associate functions due to Beardon-Minda and Baribeau-Rivard-Wegert. Basic properties of the associate functions are summarized. Then we observe that special cases of the multi-point Schwarz-Pick Lemma give the Schur’s continued fraction algorithm and several inequalities for bounded analytic functions on the unit disk.


Schur algorithm Nevanlinna-Pick interpolation Peschl’s invariant derivative Dieudonné’s Lemma 

2000 MSC

30C80 30F45 53A35 


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  1. 1.
    L. V. Ahlfors, Conformal Invariants, McGraw-Hill, New York, 1973.MATHGoogle Scholar
  2. 2.
    L. Baribeau, P. Rivard and E. Wegert, On hyperbolic divided differences and the Nevanlinna-Pick problem, Comput. Methods Funct. Theory 9 (2009), 391–405.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    A. F. Beardon and T. K. Carne, A strengthening of the Schwarz-Pick inequality, Amer. Math. Monthly 99 (1992), 216–217.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    A. F. Beardon and D. Minda, A multi-point Schwarz-Pick Lemma, J. Anal. Math. 92 (2004), 81–104.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    A. F. Beardon and D. Minda, Dieudonné points of holomorphic self-maps of regions, Comput. Methods Funct. Theory 8 (2008), 409–432.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    P. L. Duren, Univalent Functions, Springer-Verlag, 1983.Google Scholar
  7. 7.
    H. T. Kaptanoğlu, Some refined Schwarz-Pick lemmas, Michigan Math. J. 50 (2002), 649–664.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    S. Kim and T. Sugawa, Invariant differential operators associated with a conformal metric, Michigan Math. J. 55 (2007), 459–479.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    P. R. Mercer, On a strengthened Schwarz-Pick inequality, J. Math. Anal. Appl. 234 (1999), 735–739.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Z. Nehari, Conformal Mappings, McGraw-Hill, New York, 1952.Google Scholar
  11. 11.
    E. Peschl, Les invariants différentiels non holomorphes et leur rôle dans la théorie des fonctions, Rend. Sem. Mat. Messina 1 (1955), 100–108.MathSciNetGoogle Scholar
  12. 12.
    P. Rivard, Some applications of higher-order hyperbolic derivatives, preprint.Google Scholar
  13. 13.
    E. Schippers, The calculus of conformal metrics, Ann. Acad. Sci. Fenn. Math. 32 (2007), 497–521.MathSciNetMATHGoogle Scholar
  14. 14.
    I. Schur, Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, J. Reine Angew. Math. 147 (1917), 205–232, 148 (1918), 122–145; english translation in: I. Schur Methods in Operator Theory and Signal Processing, Operator Theory: Adv. and Appl. 18 (1986), Birkhäuser Verlag.Google Scholar
  15. 15.
    H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc., New York, N. Y., 1948.MATHGoogle Scholar
  16. 16.
    S. Yamashita, The Pick version of the Schwarz lemma and comparison of the Poincaré densities, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 19 (1994), 291–322.MathSciNetMATHGoogle Scholar
  17. 17.
    S. Yamashita, Norm estimates for function starlike or convex of order alpha, Hokkaido Math. J. 28 (1999), 217–230.MathSciNetMATHGoogle Scholar

Copyright information

© Heldermann  Verlag 2012

Authors and Affiliations

  • Kyung Hyun Cho
    • 1
  • Seong-A Kim
    • 2
  • Toshiyuki Sugawa
    • 3
  1. 1.Department of PhysicsPohang University of Science and TechnologyPohangKorea
  2. 2.Department of Mathematics EducationDongguk UniversityGyeongjuKorea
  3. 3.Graduate School of Information SciencesTohoku UniversityAoba-kuJapan

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