Extremal problems in classes of analytic univalent functions with quasiconformal extensions

Abstract

ABSTRACT. This work solves many of the classical extremal problems posed in the class of functions 2,,,

Extremal Problems in Classes of Analytic Univalent Functions with Quasiconformal Extensions

“…This result follows in a standard way from Theorem 2 (see [l] or [5]) or directly through an application of the variational method [8].…”

“…Presently we shall define the variation to be used and state the main conclusions of Schiffer and Schober's work. We refer the reader to [11], [10], [12], [13], [8] for details.…”

“…We will say a real valued functional A on J has a continuous variational derivative on Î if Schiffer's theorem [ll], [12], [13], [8], [10]. Suppose (ii) 3" is a normalized, compact family of K-quasiconformal mappings on ß.…”

Diameter problems for univalent functions with quasiconformal extension

“…Let The next corollary is proven by forming a Riemann sum and applying the above theorem. For an example of this, see McLeavey [6], and for another method of proof, see Kunzi [5] Next an application of corollary 1 is proven which extends corollary 7.3, page 124, in Jenkins [2]. It will be clear that one can also extend corollaries 7.1 and 7.4 in Jenkins [2].…”