Univalency conditions
conditions for univalence
Necessary and sufficient conditions for a regular (or meromorphic) function to be univalent in a domain of the complex plane (cf. Univalent function). A necessary and sufficient condition for
to be univalent in a sufficiently small neighbourhood of a point
is that
. Such (local) univalence at every point of a domain does not yet ensure univalence in the domain. For example, the function
is not univalent in the disc
, where
, although it satisfies the condition for local univalence at every point of the plane. Any property of univalent functions, and in particular any inequality satisfied by all univalent functions, is a necessary condition for univalence. The following are necessary and sufficient conditions for univalence.
Theorem 1.
Suppose that has a series expansion
![]() | (1) |
in a neighbourhood of , and let
![]() |
with constant coefficients and
. For
to be regular and univalent in
it is necessary and sufficient that for every positive integer
and all
,
, the Grunsky inequalities are satisfied:
![]() |
Similar conditions hold for the class (the class of functions
that are meromorphic and univalent in a domain
; see [2], and also Area principle).
Theorem 2.
Let the boundary of a bounded domain
be a Jordan curve. Let the function
be regular in
and continuous on the closed domain
. A necessary and sufficient condition for
to be univalent in
is that
maps
bijectively onto some closed Jordan curve.
Necessary and sufficient conditions for the function (1) on the disc to be a univalent mapping onto a convex domain, or a domain star-like or spiral-like relative to the origin, are related to theorem 2, and can be stated, respectively, in the forms
![]() |
![]() |
Many sufficient univalence conditions can be described by means of ordinary (theorem 3) or partial (theorem 4) differential equations.
Theorem 3.
A meromorphic function in the disc
is univalent in
if the Schwarzian derivative
![]() |
satisfies the inequality
![]() |
where the majorant is a non-negative continuous function satisfying the conditions: a)
does not increase in
for
; and b) the differential equation
for
has a solution
.
A special case of theorem 3 is formed by the Nehari–Pokornii univalence conditions:
![]() |
where if
and
if
.
Theorem 4.
Let be a regular function in the disc
that is continuously differentiable with respect to
,
,
, and satisfying the Löwner–Kufarev equation
![]() |
where is a regular function in
, continuous in
,
, and
. If
![]() |
where ,
is a bounded quantity as
for every
, and
is a regular non-constant function on
with expansion (1), then all functions
are univalent, including the functions
and
.
Theorem 4 implies the following special univalence conditions:
![]() |
and
![]() |
where ,
,
are real constants,
,
, and
is a regular function mapping the disc
onto a convex domain.
The univalence of the function
![]() | (2) |
is equivalent to the uniqueness of the solution of (2) in . In this sense, sufficient univalence conditions can be extended to a wide class of operator equations. For these equations, the condition
can, in particular, be generalized to a class of real mappings of domains in an
-dimensional Euclidean space.
References
[1] | N.A. Lebedev, "The area principle in the theory of univalent functions" , Moscow (1975) (In Russian) |
[2] | G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian) |
[3] | A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) (Translated from Russian) |
[4] | F.G. Avkhadiev, L.A. Aksent'ev, "The main results on sufficient conditions for an analytic function to be schlicht" Russian Math. Surveys , 30 : 4 (1975) pp. 1–64 Uspekhi Mat. Nauk , 30 : 4 (1975) pp. 3–60 |
[5] | F.D. Gakhov, "Boundary value problems" , Pergamon (1966) (Translated from Russian) |
[6] | G.G. Tumashev, M.T. Nuzhin, "Inverse boundary value problems and their applications" , Kazan' (1965) (In Russian) |
Comments
Instead of "univalence" the German word "Schlicht" is sometimes used, also in the English language literature.
References
[a1] | P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11 |
Univalency conditions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Univalency_conditions&oldid=49086