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