## The Schwarz lemmaThis volume reflects the profound impact of the century-old Schwarz lemma on contemporary complex variable theory and functional analysis. The underlying theme is the theory of intrinsic metrics on complex manifolds in finite and infinite dimensions, with reports on recent developments. In addition to full treatment of classical results and the Schwarz lemma for subharmonic and plurisubharmonic functions, the author examines Schwarz-pick systems, hyperbolic manifolds, special domains, infinitesimal metric, holomorphic curvature, the algebraic-geometric of Harris, and differential-geometric, algebraic-geometric, and fixed-point free versions of the Schwarz lemma. This self-contained text provides a synthesis of knowledge in an area of growing interest. |

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### Contents

The classical Schwarz lemma | 3 |

A Schwarz lemma for plurisubharmonic functions 20 | 18 |

The Poincare distance on the unit disc | 33 |

Copyright | |

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admissible curve algebra analytic arbitrary aRNP Aut(D Aut(P Azukawa Banach manifold Banach space biholomorphically equivalent bounded symmetric domains Caratheodory and Kobayashi Caratheodory pseudodistance Cartan chapter compact subset complex Banach manifold complex C-geodesic complex extreme point complex manifold continuous converges convex domain Corollary defined definition denote differential dimensions domain in Cn finite dimensional finite rank Finsler metric geodesic Green function Hence Hilbert space holomorphic curvature holomorphic functions holomorphic mappings holomorphic retract hyperbolic implies infinite infinitesimal Finsler pseudometric inner product irreducible isometry K-hyperbolic Kaup Kobayashi pseudodistances limsup linear mapping Math neighbourhood norm one-dimensional open unit ball plurisubharmonic functions Poincare distance Poincare metric positive integer properties proposition proved pseudoconvex pseudometric result satisfying Schwarz lemma Schwarz-Pick lemma Schwarz-Pick system sequence shows subharmonic functions submanifold subspace suppose tanh theory topology triple system unique unit ball unit disc upper semicontinuous Vesentini Vigue