A regular or meromorphic function
in a domain
of the extended complex plane
such that
whenever
,
,
that is,
is a one-to-one mapping from
into
.
The inverse function
is then also univalent. Multivalent functions (cf.
Multivalent function),
and in particular
-valent
functions, are a generalization of univalent functions.
In the study of univalent functions one of the fundamental problems
is whether there exists a univalent mapping from a given domain
onto a given domain
.
A necessary condition for the existence of such a mapping is that
and
have equal degrees of connectivity (see, for example,
[1]).
If
and
are simply-connected domains whose boundaries contain more than one
point, then this condition is also sufficient (see
Riemann theorem),
and the problem reduces to mapping a given domain onto
a disc. In this connection, a special role is
played in the theory of univalent functions on simply-connected domains by the
class
of functions
that are regular and univalent on the disc
,
normalized by the conditions
,
,
and having the expansion
In the case of multiply-connected domains, mappings of a
given multiply-connected domain onto so-called canonical domains are studied (see
Conformal mapping).
Let
be the class of functions
that are meromorphic and univalent on a domain
containing the point
,
and having an expansion
in a neighbourhood of
.
If
,
then this class is denoted by
.
The basic problems in the theory of univalent functions are the
following: 1) the study of the correspondence of boundaries under conformal mapping (see
Boundary correspondence (under conformal mapping);
Limit elements;
Attainable boundary point);
2) obtaining
univalency conditions;
and 3) the solution of various extremal problems in function theory, in
particular obtaining bounds for various functionals and for the range of values
of functionals (see below) and systems of them in some class or other.
Suppose that
is some class (set) of regular or meromorphic
functions, and suppose that a complex functional
(or system of functionals
)
is given on
.
The
range of values of the functional
(or of the system of functionals
)
on the class
is the set
of points
in
(respectively, the set in points
in
-dimensional
complex space
)
such that
.
Real-valued functionals are also considered. Any set
containing
is called a
majorant domain of the functional
(or of the system of functionals). Knowledge of the range of
values of a functional enables one to reduce the solution of
a number of extremal problems to simple problems in
analysis. For example, if the range of values
is known for the functional
,
(
fixed), then the problem of finding upper and lower bounds for
reduces to finding the points of
farthest from and closest to the point
.
The first substantial results in the theory
of univalent functions were obtained using the
area principle.
With the aid of the
outer area theorem
(1916),
L. Bieberbach
obtained precise upper and lower bounds for
and
for
(see
Distortion theorems),
gave the bound
for
and conjectured that
for
(see
Bieberbach conjecture;
Coefficient problem).
He also found the exact value of the
Koebe constant.
Bounds were found for the modulus of a function and its
derivative, as well as other bounds for the
classes of convex, star-like, typically-real, etc., functions (cf.
Convex function (of a complex variable);
Star-like function;
Typically-real function).
The
convexity radius
and the radius of
"star-likeness"
were found for a number of classes (see
Limit of star-likeness).
The basic methods of the theory of univalent functions and
some of the results obtained from them are given below.
1. The method of integral representations.
This method enables one to solve many problems in the theory
of functions quite simply, in particular extremal problems in classes
of functions that can be represented by means
of Stieltjes integrals: convex functions, close-to-convex functions, star-like
functions, typically-real functions, and functions with positive real part (see
Carathéodory class).
A variational method (see
[1])
was developed for classes of functions representable by Stieltjes integrals,
by means of which a number of extremal problems
have been solved. The method of internal variations (cf.
Internal variations, method of)
was developed for such classes.
The convex hulls of certain subclasses of
have been found (see
[3]).
In particular, it has been proved that for every star-like function
there exists a non-decreasing function
on
such that
and
See also
Integral representation of an analytic function;
Parametric representation of univalent functions;
Parametric representation method.
2. The method of boundary integration.
With the aid of this method it has been proved, in particular, that an
satisfies the inequality (see
[1])
where
and
are complete elliptic integrals (cf.
Elliptic integral).
If
is fixed
,
then this inequality determines the range of values of the functional
on the class
.
Stronger versions of the distortion theorems were
obtained, and theorems were proved on the distortion of chords in the classes
and
(see
Distortion theorems
and
[1]).
See also
Method of boundary integration;
Area principle.
3. The area method.
Let
be the class of systems of functions
mapping the disc
conformally and univalently onto pairwise disjoint (non-overlapping) domains
and normalized by the conditions
.
The following results have been obtained by means of the area theorem in the class
:
1)
If
then
this inequality generalizes an inequality previously known for real
to the class of complex
.
2)
If
,
then
For the
Bieberbach–Eilenberg functions
there follows the inequality
and conditions have been determined for equality to hold in
(4)
and
(4prm).
Using the area theorem for non-overlapping domains, bounds have
been obtained for the approximation to a regular function
on a closed multiply-connected domain by a rational
function interpolating the given function at nodes uniformly
distributed on the boundary of the domain (see
[4]).
The range of values of the
Schwarzian
has been obtained for
,
and a number of other ranges of values have been
found for classes of functions given on multiply-connected domains (see
[4],
[5]).
4. Löwner's method.
K. Löwner
himself
(1923)
found the exact bound
for functions
and exact bounds for the coefficients of the expansion of the function inverse to
,
in a neighbourhood of the point
.
In particular, an exact form of the rotation theorem in the class
was obtained by this method (see
Rotation theorems).
The following theorem was proved: For
and given
and
,
the following inequality is valid:
where
,
,
is determined by the condition
Inequality
(5)
is sharp; it implies the following sharp inequalities in the class
(
,
):
By means of distortion theorems it has been established that the
Koebe function
(
real) realizes the maximum of the linear measure of covering the circle
by the image
of the disc
under mappings by functions of class
when
.
This property of functions of class
implies bounds for the area
of the domain
,
bounds for the average modulus of a function, and other bounds in the class
;
these are asymptotically sharp as
(see
[1]).
A convenient reduction of extremal problems on
and some of its subclasses to certain extremal
problems on a simpler class has been proposed (see
Carathéodory class).
This turns out to be applicable to the solution of several extremal problems,
in particular to finding the range of values of the system of functionals
(here
,
,
is fixed) for
(see
[6]).
Löwner's method has been successfully applied to investigate the properties
of level curves and to solve extremal problems in the subclass
of bounded functions
:
,
(see
[6]).
See also
Löwner equation;
Löwner method;
Parametric representation method.
5. Variational methods.
Boundary and internal variations in the solution of extremal problems
lead to differential equations for the boundaries of the extremal
domains and for extremal functions, respectively. As a rule,
the left-hand side of these equations is a
quadratic differential.
Various qualitative characteristics of the functions realizing
the extremum can be obtained by investigating the
properties of the corresponding quadratic differentials. In particular, it
turns out that for a large number of extremal problems in the class
(and in other classes), the extremal function maps the disc
onto the whole plane with a finite number
of analytic slits. Sometimes the differential equation for the extremal
function can be integrated, and one obtains the extremal quantity
and all extremal functions in the problem considered. More often
one only obtains one or a few equations for the
extremal quantity. Some results obtained by variational methods are listed below.
Suppose that
,
that
,
,
,
do not belong to the image of the domain
under the mapping
,
and that
It has been proved that
with equality only for
where
is real (see
[1]).
It has been proved (see
[1])
that the range of values of the functional
for
,
where
are given numbers not all zero and
are given points in
,
is the disc
The problem has been investigated of extremizing
,
,
in the class
of functions
that are regular and univalent in the disc
and do not take on given values
in
(see
[1]).
The special case
is the problem of determining a
continuum of least capacity
(for a consideration of this problem and its generalizations see
[7]).
Various problems for non-overlapping domains have been investigated by
a variational method. Thus, the problem of maximizing the product
in the class
has been considered (see
[1]).
A precise bound has been obtained for the product
where
are any given positive numbers, for
and 3 (see
[1]).
This problem is equivalent to the problem of finding the range
of the system of functionals
in the class
.
See also
Variational principles (in complex function theory);
Variation of a univalent function;
Internal variations, method of;
Boundary variation, method of;
Variation-parametric method.
6. The method of the extremal metric.
In the solution of extremal problems by the method of the extremal
metric, a fundamental role is played, as a rule,
by the metric generated by a certain quadratic differential
.
This is the same quadratic differential that arises in the solution of
the problem by the variational method. As an example, two
results obtained by this method are given below (see
[1],
[7]–[9]).
By means of the
general coefficient theorem,
J.A. Jenkins
(1960)
has solved the problem of the range of values of the functional
for fixed
in the disc
in the class
of functions in
with real coefficients
.
In the classes
and
,
where
is the class of functions
,
,
,
that are meromorphic and univalent in the disc
,
he clarified the influence of the vanishing of a certain number
of the initial coefficients on the growth of the subsequent ones.
A supplement to the general coefficient theorem has
been given in the case when the differential
has no poles of order higher than one; in addition, by means
of the extremal-metric approach, very general theorems have been established on the
covering of curves under a univalent conformal mapping of simply-
and doubly-connected domains, including, in particular, a refinement of the
result on covering of intervals for functions meromorphic and univalent
on the disc, and an analogous result for a circular annulus (see
[1]).
See also
Grötzsch principle;
Grötzsch theorems;
Strip method (analytic functions);
Quadratic differential;
Bieberbach–Eilenberg functions;
Extremal metric, method of the.
7. The method of symmetrization.
Several complicated extremal problems not lending themselves to solution by other
methods have been solved by this method, often in conjunction with
others. For example, the following problems are of this kind (see
[1],
[7]–[10]).
For functions
in the class
,
a sharp upper bound has been found for the set of points of the circle
,
,
not belonging to the image of the disc
under the mapping
.
In conjunction with the method of the extremal metric,
a sharp upper bound has been found for
for fixed
,
,
for
with given
,
;
the inequalities
(6)
have been generalized and extended
to the class of functions that are
-valent
in the mean on the circle:
(see
Multivalent function).
By the method of symmetrization it has been proved that if
is a convex and non-decreasing function on
,
then for
and
,
where
(see
[11]).
If equality holds for some
,
,
and for some strictly convex function
,
then
where
is real.
See
[12],
[13]
for applications of the method of symmetrization to multiply-connected domains. See also
Symmetrization method.