CALCULATION OF EXTREMUM PROBLEMS FOR UNIVALENT FUNCTIONS

Abstract

Let $S$ be the usual class of univalent functions in {${|z|~<~1}$} normalized by $f(z)~=~z~+$ $sum from i=2 to {inf}$ $a sub i z sup i$ and $V sub n$ the coefficient region of $S$. It is well known that $f$ corresponds to a boundary point of $V sub n$ if and only if $f$ satisfies a quadratic equation of the form $Q(w)dw sup 2~=~R(z)dz sup 2$ called Schiffers equation that maps {${|z|~<~1}$} onto a slit domain. We treat the following problems numerically for $V sub 4$: 1. Given $Q$ find $R$ and $f$. 2. Find the function that maximizes $Re~e sup i sup phi$ $a sub 4$ with theconstraint that $a sub 2$ and $a sub 3$ are some given complex numbers in $V sub 3$. In this case Schiffers equation is a sufficient condition for $f$ to be extremal. The critical trajectories of $Q(w)dw sup 2$ and $R(z)dz sup 2$ are in each case displayed graphically for some particular examples.