Abstract
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.
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References
L. V. Ahlfors, Conformal Invariants, McGraw-Hill, New York, 1973.
L. Baribeau, P. Rivard and E. Wegert, On hyperbolic divided differences and the Nevanlinna-Pick problem, Comput. Methods Funct. Theory 9 (2009), 391–405.
A. F. Beardon and T. K. Carne, A strengthening of the Schwarz-Pick inequality, Amer. Math. Monthly 99 (1992), 216–217.
A. F. Beardon and D. Minda, A multi-point Schwarz-Pick Lemma, J. Anal. Math. 92 (2004), 81–104.
A. F. Beardon and D. Minda, Dieudonné points of holomorphic self-maps of regions, Comput. Methods Funct. Theory 8 (2008), 409–432.
P. L. Duren, Univalent Functions, Springer-Verlag, 1983.
H. T. Kaptanoğlu, Some refined Schwarz-Pick lemmas, Michigan Math. J. 50 (2002), 649–664.
S. Kim and T. Sugawa, Invariant differential operators associated with a conformal metric, Michigan Math. J. 55 (2007), 459–479.
P. R. Mercer, On a strengthened Schwarz-Pick inequality, J. Math. Anal. Appl. 234 (1999), 735–739.
Z. Nehari, Conformal Mappings, McGraw-Hill, New York, 1952.
E. Peschl, Les invariants différentiels non holomorphes et leur rôle dans la théorie des fonctions, Rend. Sem. Mat. Messina 1 (1955), 100–108.
P. Rivard, Some applications of higher-order hyperbolic derivatives, preprint.
E. Schippers, The calculus of conformal metrics, Ann. Acad. Sci. Fenn. Math. 32 (2007), 497–521.
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.
H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc., New York, N. Y., 1948.
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.
S. Yamashita, Norm estimates for function starlike or convex of order alpha, Hokkaido Math. J. 28 (1999), 217–230.
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The third author was supported in part by JSPS Grant-in-Aid for Scientific Research (B), 17340039 and for Exploratory Research, 19654027.
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Cho, K.H., Kim, SA. & Sugawa, T. On a Multi-Point Schwarz-Pick Lemma. Comput. Methods Funct. Theory 12, 483–499 (2012). https://doi.org/10.1007/BF03321839
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DOI: https://doi.org/10.1007/BF03321839