Schwarz-Pick Type Inequalities

The aim of the present book is a uni?ed representation of some recent results in geometric function theory together with a consideration of their historical sources. These results are concerned with functions f, holomorphic or meromorphic in a domain ? in the extended complex planeC. The only additional condition we impose on these functions is the condition that the range f(?) is contained in a given domain ??C.Thisfactwillbedenotedby f? A(?,?). We shall describe (n) how one may get estimates for the derivatives|f (z )|,n?N,f ? A(?,?), 0 dependent on the position of z in ? and f(z)in?. 0 0 1.1 Historical remarks The beginning of this program may be found in the famous article [125] of G. Pick. There, he discusses estimates for the MacLaurin coe?cients of functions with positive real part in the unit disc found by C. Carath¿ eodory in [52]. Pick tells his readers that he wants to generalize Carath¿ eodory¿s estimates such that the special role of the expansion point at the origin is no longer important. For the convenience of our readers we quote this sentence in the original language: Durch lineare Transformation von z oder, wie man sagen darf, durch kreis- ometrische Verallgemeinerung, kann man die Sonderstellung des Wertes z=0 wegscha?en, so daß sich Relationen fur ¿ die Di?erentialquotienten von w an - liebiger Stelle ergeben. The ?rst great success of this program was G. Pick¿s theorem, as it is called by Carath¿ eodory himself, compare [54], vol II, §286¿289.