# 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.

## Contents

### 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)