Schwarz' lemma in invariant form
The following generalization of the Schwarz lemma. Let be a bounded regular analytic function in the unit disc , for . Then for any points and in the non-Euclidean distance of their images and does not exceed the non-Euclidean distance , i.e.
One also has the inequality
between the elements of non-Euclidean length (the differential form of Pick's theorem or the Schwarz lemma). Equality applies in (1) and (2) only if is a Möbius function that maps onto itself (cf. Fractional-linear mapping).
The non-Euclidean, or hyperbolic, distance is the distance in Lobachevskii geometry between and when is the Lobachevskii plane and arcs of circles serve as Lobachevskii straight lines, these being orthogonal to the unit circle (Poincaré's model), and
where is the cross ratio between the points and and the points of intersection and of the Lobachevskii straight line passing through and with the unit circle (see Fig.).
The non-Euclidean length of the image of any rectifiable curve under the mapping does not exceed the non-Euclidean length of .
The theorem was established by G. Pick ; a far-reaching generalization of it is provided by the principle of the hyperbolic metric (cf. Hyperbolic metric, principle of the). In geometric function theory these theorems provide bounds for various functionals related to mapping functions , .
|||G. Pick, "Ueber eine Eigenschaft der konformen Abbildung kreisförmiger Bereiche" Math. Ann. , 77 (1916) pp. 1–6|
|||G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)|
|||C. Carathéodory, "Conformal representation" , Cambridge Univ. Press (1952)|
|||J.B. Garnett, "Bounded analytic functions" , Acad. Press (1981)|
|[a1]||L.V. Ahlfors, "Conformal invariants. Topics in geometric function theory" , McGraw-Hill (1973)|
|[a2]||S. Lang, "Introduction to complex hyperbolic spaces" , Springer (1987)|
Pick theorem. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Pick_theorem&oldid=12628