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''conditions for univalence''

Necessary and sufficient conditions for a regular (or meromorphic) function to be univalent in a domain of the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u0956001.png" /> (cf. [[Univalent function|Univalent function]]). A necessary and sufficient condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u0956002.png" /> to be univalent in a sufficiently small neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u0956003.png" /> is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u0956004.png" />. Such (local) univalence at every point of a domain does not yet ensure univalence in the domain. For example, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u0956005.png" /> is not univalent in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u0956006.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u0956007.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u0956008.png" /> has a series expansion

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u0956009.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560010.png" />, and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560011.png" /></td> </tr></table>

with constant coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560013.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560014.png" /> to be regular and univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560015.png" /> it is necessary and sufficient that for every positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560016.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560018.png" />, the Grunsky inequalities are satisfied:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560019.png" /></td> </tr></table>

Similar conditions hold for the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560021.png" /> (the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560022.png" /> that are meromorphic and univalent in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560023.png" />; see [[#References|[2]]], and also [[Area principle|Area principle]]).

===Theorem 2.===
Let the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560024.png" /> of a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560025.png" /> be a Jordan curve. Let the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560026.png" /> be regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560027.png" /> and continuous on the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560028.png" />. A necessary and sufficient condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560029.png" /> to be univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560030.png" /> is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560031.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560032.png" /> bijectively onto some closed Jordan curve.

Necessary and sufficient conditions for the function (1) on the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560033.png" /> 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560034.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560035.png" /></td> </tr></table>

Many sufficient univalence conditions can be described by means of ordinary (theorem 3) or partial (theorem 4) differential equations.

===Theorem 3.===
A meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560036.png" /> in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560037.png" /> is univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560038.png" /> if the Schwarzian derivative

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560039.png" /></td> </tr></table>

satisfies the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560040.png" /></td> </tr></table>

where the majorant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560041.png" /> is a non-negative continuous function satisfying the conditions: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560042.png" /> does not increase in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560043.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560044.png" />; and b) the differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560045.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560046.png" /> has a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560047.png" />.

A special case of theorem 3 is formed by the Nehari–Pokornii univalence conditions:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560048.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560049.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560051.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560052.png" />.

===Theorem 4.===
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560053.png" /> be a regular function in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560054.png" /> that is continuously differentiable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560057.png" />, and satisfying the Löwner–Kufarev equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560058.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560059.png" /> is a regular function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560060.png" />, continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560062.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560063.png" />. If

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560064.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560066.png" /> is a bounded quantity as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560067.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560068.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560069.png" /> is a regular non-constant function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560070.png" /> with expansion (1), then all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560071.png" /> are univalent, including the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560073.png" />.

Theorem 4 implies the following special univalence conditions:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560074.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560075.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560078.png" /> are real constants, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560080.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560081.png" /> is a regular function mapping the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560082.png" /> onto a convex domain.

The univalence of the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560083.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

is equivalent to the uniqueness of the solution of (2) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560084.png" />. In this sense, sufficient univalence conditions can be extended to a wide class of operator equations. For these equations, the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560085.png" /> can, in particular, be generalized to a class of real mappings of domains in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095600/u09560086.png" />-dimensional Euclidean space.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.A. Lebedev,   "The area principle in the theory of univalent functions" , Moscow  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Goluzin,   "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  F.D. Gakhov,   "Boundary value problems" , Pergamon  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.G. Tumashev,   M.T. Nuzhin,   "Inverse boundary value problems and their applications" , Kazan'  (1965)  (In Russian)</TD></TR></table>



====Comments====
Instead of  "univalence"  the German word  "Schlicht"  is sometimes used, also in the English language literature.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Duren,   "Univalent functions" , Springer  (1983)  pp. Sect. 10.11</TD></TR></table>