A Variational Theory for Biunivalent Holomorphic Functions

Biunivalent holomorphic functions form an interesting class in geometric function theory and are connected with special functions and solutions of complex differential equations. This class has been investigated by many authors, mainly to find the coefficient estimates. The assumption of biunivalence is rigid; this rigidity means that, for example, only the initial Taylor coefficients have been estimated. The aim of this paper is to develop a variational technique for biunivalent functions, which provides a power tool for solving the general extremal problems on the classes of such functions. It involves quasiconformal analysis.