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.

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

$$ \tag{1 } f ( z) = z + a _ {2} z ^ {2} + \dots + a _ {n} z ^ {n} + \dots $$

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

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

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:

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

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.

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

$$ \mathop{\rm Re} \left ( z \frac{f ^ { \prime\prime } ( z) }{f ^ { \prime } ( z) } \right ) + 1 \geq 0 ,\ \mathop{\rm Re} \left ( z \frac{f ^ { \prime } ( z) }{f ( z) } \right ) \geq 0 , $$

$$ \mathop{\rm Re} \left ( e ^ {i \gamma } z \frac{f ^ { \prime } ( z) }{f ( z) } \right ) \geq 0 . $$

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

Theorem 3.

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

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

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:

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

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.

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

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

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:

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

and

$$ \mathop{\rm Re} \left [ e ^ {i \gamma } \left ( \frac{f ( z) }{z} \right ) ^ {\alpha + i \beta - 1 } \frac{f ^ { \prime } ( z) }{\phi ^ {\prime \alpha } ( z) } \right ] \geq 0 , $$

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

$$ \tag{2 } w = f ( z) $$

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

[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