Univalent functions that are local extrema of two real functionals. (English) Zbl 0829.30011

Summary: The class \(S\) consists of all functions \(f(z) = z + c_2 z^2 + \cdots\) that are regular and univalent in the unit disc. Let the functionals \(F(c_2, \ldots, c_n)\) and \(\Phi (c_2, \ldots, c_m)\) have nonvanishing gradient in domains containing sets of the type \(\{|c_2 |\leq 2, \ldots, |c_r |\leq r\}\). A function \(f_0 \in S\) is found for which the functionals \(F\) and \(\Phi\) attain a local extremum.


30C70 Extremal problems for conformal and quasiconformal mappings, variational methods
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)