A method of studying univalent functions (cf.
Univalent function),
based on the study of the
variation of a function
that is univalent in a domain of the
-plane,
the variation of the function being determined by the
appropriate variations of the boundary of the image of this domain.
The
fundamental lemma of the method of boundary variation.
Let
be a domain in the
-plane
and let the complement
of
in the extended plane consist of a number of continua. Let
be a continuum in
and let there exist on
an analytic function
such that for any point
and for any function
that can be represented as
and that is univalent in
,
the inequality
be valid, and suppose that the estimate of the residual
term in
(*)
is uniform in all closed subdomains of
.
will then be an analytic curve that may
parametrically be represented by means of the function
of the real parameter
.
This parameter may be so chosen that
satisfies the differential equation
This result makes the important role played by quadratic differentials (cf.
Quadratic differential)
in the solution of extremal problems in the theory of univalent functions clear, since
proves to be a meromorphic function in many applications.
In certain cases it follows from the conditions
of the problem that the appropriate poles of
belong to the boundary of the extremal domain, and it
is shown by the fundamental lemma of the method of
boundary variation that the boundary of this domain belongs to
the union of the closures of the critical trajectories of the quadratic differential
In a number of extremal problems, the fundamental lemma
not only yields qualitative results, but also gives sufficient information
for the determination of the boundary of the extremal domain,
and hence for the complete solution of the problem.
The following results were obtained by means of the
method of boundary variation: Qualitative results in the
coefficient problem
for the class
;
in the problem of the maximum of the
-th
diameter in a family of continua
of a given capacity; the solution of a number
of extremal problems of univalent conformal mappings of doubly-connected domains;
distortion theorems
for multiply-connected domains, which at the same
prove existence theorems of univalent conformal mappings
of a given multiply-connected domain onto canonical domains, etc.