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.
Description
Keywords
Computer Science