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 derivativesf (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.
