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 $ \mathbf C $ (cf. Univalent function). A necessary and sufficient condition for $ f ( z) $ to be univalent in a sufficiently small neighbourhood of a point $ a $ is that $ f ^ { \prime } ( a) \neq 0 $. Such (local) univalence at every point of a domain does not yet ensure univalence in the domain. For example, the function $ e ^ {z} $ is not univalent in the disc $ | z | \leq R $, where $ R > \pi $, 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.[edit]

Suppose that $ f ( z) $ has a series expansion

$$\begin{array}{}\text{(1)}& f(z)=z+a_2{z}^2+\cdots +a_n{z}^n+\dots \end{array}$$

in a neighbourhood of $ z = 0 $, and let

$$\ln \frac{f(t)-f(z)}{t-z}=\sum _{p,q=0}^{\mathrm{\infty }}{\omega }_{p,q}{t}^p{z}^q$$

with constant coefficients $ a _ {k} $ and $ \omega _ {p,q} $. For $ f ( z) $ to be regular and univalent in $ E = \{ {z } : {| z | < 1 } \} $ it is necessary and sufficient that for every positive integer $ N $ and all $ x _ {p} $, $ p = 1 \dots N $, the Grunsky inequalities are satisfied:

$$\left|\sum _{p,q=1}^N{\omega }_{p,q}{x}_p{x}_q\right|\le \sum _{p=1}^N\frac{1}{p}|x_p{|}^2.$$

Similar conditions hold for the class $ \Sigma ( B) $ (the class of functions $ F ( \zeta ) = \zeta + c _ {0} + c _ {1} / \zeta + \dots $ that are meromorphic and univalent in a domain $ B \ni \infty $; see [2], and also Area principle).

Theorem 2.[edit]

Let the boundary $ l $ of a bounded domain $ D $ be a Jordan curve. Let the function $ f ( z) $ be regular in $ D $ and continuous on the closed domain $ \overline{D} $. A necessary and sufficient condition for $ f ( z) $ to be univalent in $ \overline{D} $ is that $ f $ maps $ l $ bijectively onto some closed Jordan curve.

Necessary and sufficient conditions for the function (1) on the disc $ E $ 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

$$\mathrm{Re}\left(z\frac{f^{\prime \prime }(z)}{f^{\prime }(z)}\right)+1\ge 0,\mathrm{Re}\left(z\frac{f^{\prime }(z)}{f(z)}\right)\ge 0,$$

$$\mathrm{Re}\left(e^{i\gamma }z\frac{f^{\prime }(z)}{f(z)}\right)\ge 0.$$

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

Theorem 3.[edit]

A meromorphic function $ f ( z) $ in the disc $ E $ is univalent in $ E $ if the Schwarzian derivative

$$\{f,z\}={\left[\frac{f^{\prime \prime }(z)}{f^{\prime }(z)}\right]}^{\prime }-\frac{1}{2}{\left[\frac{f^{\prime \prime }(z)}{f^{\prime }(z)}\right]}^2$$

satisfies the inequality

$$|\{f,z\}|\le 2S(|z|),|z|<1,$$

where the majorant $ S ( r) $ is a non-negative continuous function satisfying the conditions: a) $ S ( r) ( 1 - r ^ {2} ) ^ {2} $ does not increase in $ r $ for $ 0 < r < 1 $; and b) the differential equation $ y ^ {\prime\prime} + S ( | t | ) y = 0 $ for $ - 1 < t < 1 $ has a solution $ y _ {0} ( t) > 0 $.

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

$$|\{f,z\}|\le \frac{C(\mu )}{(1-|z{|}^2{)}^{\mu }},$$

where $ C ( \mu ) = 2 ^ {3 \mu - 1 } \pi ^ {2 ( 1 - \mu ) } $ if $ 0 \leq \mu \leq 1 $ and $ = 2 ^ {3 - \mu } $ if $ 1 \leq \mu \leq 2 $.

Theorem 4.[edit]

Let $ f ( z , t ) $ be a regular function in the disc $ E $ that is continuously differentiable with respect to $ t $, $ 0 \leq t < \infty $, $ f ( 0 , t ) = 0 $, and satisfying the Löwner–Kufarev equation

$$\frac{\partial f}{\partial t}=zh(z,t)\frac{\partial f}{\partial z},0<t<\mathrm{\infty },z\in E,$$

where $ h ( z , t ) $ is a regular function in $ E $, continuous in $ t $, $ 0 \leq t < \infty $, and $ \mathop{\rm Re} h ( z , t ) \geq 0 $. If

$$f(z,t)=a_0(t)f(z)+O(1),$$

where $ \lim\limits _ {t \rightarrow \infty } a _ {0} ( t) = \infty $, $ O ( 1) $ is a bounded quantity as $ t \rightarrow \infty $ for every $ z \in E $, and $ f ( z) $ is a regular non-constant function on $ E $ with expansion (1), then all functions $ f ( z , t ) $ are univalent, including the functions $ f ( z , 0 ) $ and $ f ( z) $.

Theorem 4 implies the following special univalence conditions:

$$\left|z\frac{f^{\prime \prime }(z)}{f^{\prime }(z)}\right|\le \frac{1}{1-|z{|}^2}$$

and

$$\mathrm{Re}\left[e^{i\gamma }{\left(\frac{f(z)}{z}\right)}^{\alpha +i\beta -1}\frac{f^{\prime }(z)}{{\varphi }^{\prime \alpha }(z)}\right]\ge 0,$$

where $ \alpha $, $ \beta $, $ \gamma $ are real constants, $ \alpha > 0 $, $ | \gamma | < \pi / 2 $, and $ \phi ( z) $ is a regular function mapping the disc $ E $ onto a convex domain.

The univalence of the function

$$\begin{array}{}\text{(2)}& w=f(z)\end{array}$$

is equivalent to the uniqueness of the solution of (2) in $ z $. In this sense, sufficient univalence conditions can be extended to a wide class of operator equations. For these equations, the condition $ \mathop{\rm Re} [ e ^ {i \gamma } f ^ { \prime } ( z) ] \geq 0 $ can, in particular, be generalized to a class of real mappings of domains in an $ n $-dimensional Euclidean space.

References[edit]

Comments[edit]

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

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