The General Coefficient Theorem

Abstract

Teichmüller [188] enunciated the principle that the solution of a certain type of extremal problem in Geometric Function Theory is in general associated with a quadratic differential. If in the problem a point is assumed to be fixed without further requirement the quadratic differential will have a simple pole there. If in addition the functions treated in the problem are required to have at the point, in terms of suitably assigned local uniformizing parameters, fixed values for their first n derivatives, the quadratic differential will have a pole of order n + 1 there. More generally, the highest derivative occurring may not be required to be fixed but some condition on its region of variation may be desired. Teichmüller was led to this principle by abstraction from the numerous results of Grötzsch [61–78] and by his consideraions on quasiconformal mappings. However he never gave anything in the nature of an explicit general result embodying this principle.