Pavel G. Todorov
Taylor Series of the Pick Function and the Loewner Variation within the Class \$\boldsymbol S\$ and Applications
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 54, 91-99 (2000)
MSC:
30B10 Power series (including lacunary series)
30C50 Coefficient problems for univalent and multivalent functions
30C70 Extremal problems for conformal and quasiconformal mappings, variational methods
33C05 Classical hypergeometric functions, \$_2F_1\$
Abstract: We find the unknown Taylor series of the Pick function
(4) (or (8) and the Loewner variation (5). As an application we prove the local Bieberbach
conjecture for the Loewner variation (5) in a simple
way (see Corollaries 3.2 and 3.3 and their consequences).
Keywords: Pick function, Loewner variation, Gauss hypergeometric function, Taylor series, local Bieberbach conjecture, Bombieri proof in the class \$S\$, our proof for the Loewner variation within the class \$S\$