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Schwarz-Pick Type Inequalities (Paperback, 2009 ed.)
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Schwarz-Pick Type Inequalities (Paperback, 2009 ed.)
Series: Frontiers in Mathematics
Expected to ship within 10 - 15 working days
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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.
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