| Local Mininmality Results Related To The Bloch And Landau Constants (2007) | |||||||||||||
Abstract | |||||||||||||
| This paper has two goals. One is to present some modest evidence in favor of the Ahlfors-Grunsky and Rademacher conjectures. The other is to highlight some methods which are used in the proofs of our results, notably second variations of quasiconformal mappings. A number of authors have used this method to attack various problems of complex analysis, but we feel that it potentially has many more uses, especially to concrete extremal problems like those of finding the Bloch and Landau constants. To the newcomer, second variations can look scary. Accordingly, we have supplied semidetailed derivations of the main formulas needed for our proofs, even though it would have posssible to cite most of them from the published literature. To explain our results, we first examine the conjectured extremal functions. They are associated with hexagonal lattices. In this paper, the "lattice generated by z 1 ; z 2 ; z 3 ;" where z 1 ; z 2 ; z 3 2 C ; | |||||||||||||
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