A regular or meromorphic function in a domain of the extended complex plane such that whenever , , that is, is a one-to-one mapping from into . The inverse function is then also univalent. Multivalent functions (cf. Multivalent function), and in particular -valent functions, are a generalization of univalent functions.
In the study of univalent functions one of the fundamental problems is whether there exists a univalent mapping from a given domain onto a given domain . A necessary condition for the existence of such a mapping is that and have equal degrees of connectivity (see, for example, ). If and are simply-connected domains whose boundaries contain more than one point, then this condition is also sufficient (see Riemann theorem), and the problem reduces to mapping a given domain onto a disc. In this connection, a special role is played in the theory of univalent functions on simply-connected domains by the class of functions that are regular and univalent on the disc , normalized by the conditions , , and having the expansion
In the case of multiply-connected domains, mappings of a given multiply-connected domain onto so-called canonical domains are studied (see Conformal mapping). Let be the class of functions that are meromorphic and univalent on a domain containing the point , and having an expansion
in a neighbourhood of . If , then this class is denoted by .
The basic problems in the theory of univalent functions are the following: 1) the study of the correspondence of boundaries under conformal mapping (see Boundary correspondence (under conformal mapping); Limit elements; Attainable boundary point); 2) obtaining univalency conditions; and 3) the solution of various extremal problems in function theory, in particular obtaining bounds for various functionals and for the range of values of functionals (see below) and systems of them in some class or other.
Suppose that is some class (set) of regular or meromorphic functions, and suppose that a complex functional (or system of functionals ) is given on . The range of values of the functional (or of the system of functionals ) on the class is the set of points in (respectively, the set in points in -dimensional complex space ) such that . Real-valued functionals are also considered. Any set containing is called a majorant domain of the functional (or of the system of functionals). Knowledge of the range of values of a functional enables one to reduce the solution of a number of extremal problems to simple problems in analysis. For example, if the range of values is known for the functional , ( fixed), then the problem of finding upper and lower bounds for reduces to finding the points of farthest from and closest to the point .
The first substantial results in the theory of univalent functions were obtained using the area principle. With the aid of the outer area theorem (1916), L. Bieberbach obtained precise upper and lower bounds for and for (see Distortion theorems), gave the bound for and conjectured that for (see Bieberbach conjecture; Coefficient problem). He also found the exact value of the Koebe constant. Bounds were found for the modulus of a function and its derivative, as well as other bounds for the classes of convex, star-like, typically-real, etc., functions (cf. Convex function (of a complex variable); Star-like function; Typically-real function). The convexity radius and the radius of "star-likeness" were found for a number of classes (see Limit of star-likeness).
The basic methods of the theory of univalent functions and some of the results obtained from them are given below.
1. The method of integral representations.
This method enables one to solve many problems in the theory of functions quite simply, in particular extremal problems in classes of functions that can be represented by means of Stieltjes integrals: convex functions, close-to-convex functions, star-like functions, typically-real functions, and functions with positive real part (see Carathéodory class). A variational method (see ) was developed for classes of functions representable by Stieltjes integrals, by means of which a number of extremal problems have been solved. The method of internal variations (cf. Internal variations, method of) was developed for such classes.
The convex hulls of certain subclasses of have been found (see ). In particular, it has been proved that for every star-like function there exists a non-decreasing function on such that and
2. The method of boundary integration.
With the aid of this method it has been proved, in particular, that an satisfies the inequality (see )
where and are complete elliptic integrals (cf. Elliptic integral). If is fixed , then this inequality determines the range of values of the functional on the class . Stronger versions of the distortion theorems were obtained, and theorems were proved on the distortion of chords in the classes and (see Distortion theorems and ).
3. The area method.
Let be the class of systems of functions mapping the disc conformally and univalently onto pairwise disjoint (non-overlapping) domains and normalized by the conditions . The following results have been obtained by means of the area theorem in the class :
this inequality generalizes an inequality previously known for real to the class of complex .
2) If , then
For the Bieberbach–Eilenberg functions
there follows the inequality
and conditions have been determined for equality to hold in (4) and (4prm).
Using the area theorem for non-overlapping domains, bounds have been obtained for the approximation to a regular function on a closed multiply-connected domain by a rational function interpolating the given function at nodes uniformly distributed on the boundary of the domain (see ). The range of values of the Schwarzian
4. Löwner's method.
K. Löwner himself (1923) found the exact bound for functions and exact bounds for the coefficients of the expansion of the function inverse to , in a neighbourhood of the point . In particular, an exact form of the rotation theorem in the class was obtained by this method (see Rotation theorems). The following theorem was proved: For and given and , the following inequality is valid:
where , , is determined by the condition
Inequality (5) is sharp; it implies the following sharp inequalities in the class (, ):
By means of distortion theorems it has been established that the Koebe function
( real) realizes the maximum of the linear measure of covering the circle by the image of the disc under mappings by functions of class when . This property of functions of class implies bounds for the area of the domain , bounds for the average modulus of a function, and other bounds in the class ; these are asymptotically sharp as (see ).
A convenient reduction of extremal problems on and some of its subclasses to certain extremal problems on a simpler class has been proposed (see Carathéodory class). This turns out to be applicable to the solution of several extremal problems, in particular to finding the range of values of the system of functionals (here , , is fixed) for (see ).
Löwner's method has been successfully applied to investigate the properties of level curves and to solve extremal problems in the subclass of bounded functions : , (see ).
5. Variational methods.
Boundary and internal variations in the solution of extremal problems lead to differential equations for the boundaries of the extremal domains and for extremal functions, respectively. As a rule, the left-hand side of these equations is a quadratic differential. Various qualitative characteristics of the functions realizing the extremum can be obtained by investigating the properties of the corresponding quadratic differentials. In particular, it turns out that for a large number of extremal problems in the class (and in other classes), the extremal function maps the disc onto the whole plane with a finite number of analytic slits. Sometimes the differential equation for the extremal function can be integrated, and one obtains the extremal quantity and all extremal functions in the problem considered. More often one only obtains one or a few equations for the extremal quantity. Some results obtained by variational methods are listed below.
Suppose that , that , , , do not belong to the image of the domain under the mapping , and that
It has been proved that
with equality only for
where is real (see ).
It has been proved (see ) that the range of values of the functional
for , where are given numbers not all zero and are given points in , is the disc
The problem has been investigated of extremizing , , in the class of functions that are regular and univalent in the disc and do not take on given values in (see ). The special case is the problem of determining a continuum of least capacity (for a consideration of this problem and its generalizations see ).
Various problems for non-overlapping domains have been investigated by a variational method. Thus, the problem of maximizing the product
in the class has been considered (see ). A precise bound has been obtained for the product
where are any given positive numbers, for and 3 (see ). This problem is equivalent to the problem of finding the range of the system of functionals in the class .
6. The method of the extremal metric.
In the solution of extremal problems by the method of the extremal metric, a fundamental role is played, as a rule, by the metric generated by a certain quadratic differential . This is the same quadratic differential that arises in the solution of the problem by the variational method. As an example, two results obtained by this method are given below (see , –).
By means of the general coefficient theorem, J.A. Jenkins (1960) has solved the problem of the range of values of the functional for fixed in the disc in the class of functions in with real coefficients . In the classes and , where is the class of functions , , , that are meromorphic and univalent in the disc , he clarified the influence of the vanishing of a certain number of the initial coefficients on the growth of the subsequent ones.
A supplement to the general coefficient theorem has been given in the case when the differential has no poles of order higher than one; in addition, by means of the extremal-metric approach, very general theorems have been established on the covering of curves under a univalent conformal mapping of simply- and doubly-connected domains, including, in particular, a refinement of the result on covering of intervals for functions meromorphic and univalent on the disc, and an analogous result for a circular annulus (see ).
7. The method of symmetrization.
Several complicated extremal problems not lending themselves to solution by other methods have been solved by this method, often in conjunction with others. For example, the following problems are of this kind (see , –). For functions in the class , a sharp upper bound has been found for the set of points of the circle , , not belonging to the image of the disc under the mapping . In conjunction with the method of the extremal metric, a sharp upper bound has been found for for fixed , , for
with given , ; the inequalities (6) have been generalized and extended to the class of functions that are -valent in the mean on the circle:
(see Multivalent function).
By the method of symmetrization it has been proved that if is a convex and non-decreasing function on , then for and ,
where (see ). If equality holds for some , , and for some strictly convex function , then
where is real.
|||G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)|
|||N.A. Lebedev, I.A. Aleksandrov, "On the variational method in classes of functions representable by means of Stieltjes integrals" Proc. Steklov Inst. Math. , 94 (1969) pp. 91–104 Trudy Mat. Inst. Steklov. , 94 (1968) pp. 79–89|
|||L. Brickman, T.H. MacGregor, D.R. Wilken, "Convex hulls of some classical families of univalent functions" Trans. Amer. Math. Soc. , 156 (1971) pp. 91–107|
|||N.A. Lebedev, "The area principle in the theory of univalent functions" , Moscow (1975) (In Russian)|
|||I.M. Milin, "Univalent functions and orthonormal systems" , Transl. Math. Monogr. , 49 , Amer. Math. Soc. (1977) (Translated from Russian)|
|||I.A. Aleksandrov, "Parametric extensions in the theory of univalent functions" , Moscow (1976) (In Russian)|
|||G.V. Kuz'mina, "Moduli of families of curves and quadratic differentials" Proc. Steklov Inst. Math. , 139 (1982) Trudy Mat. Inst. Steklov. , 139 (1980) pp. 1–241|
|||J.A. Jenkins, "Univalent functions and conformal mappings" , Springer (1958)|
|||C. Pommerenke, "Univalent functions" , Vandenhoeck & Ruprecht (1975)|
|||W.K. Hayman, "Multivalent functions" , Cambridge Univ. Press (1958)|
|||A. Baernstein, "Integral means, univalent functions and circular symmetrization" Acta Math. , 133 (1974) pp. 139–169|
|||O.P. Mityuk, "The symmetrization principle for multiply-connected domains and certain of its applications" Ukrain. Mat. Zh. , 17 : 4 (1965) pp. 46–54 (In Russian)|
|||I.P. Mityuk, "The symmetrization principle for an annulus and certain of its applications" Sibirsk. Mat. Zh. , 6 : 6 (1965) pp. 1282–1291 (In Russian)|
Instead of "univalent function" the phrase "Schlicht function" is sometimes used.
|[a1]||P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11|
|[a2]||A.W. Goodman, "Univalent functions" , 1–2 , Mariner (1983)|
Univalent function. N.A. Lebedev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Univalent_function&oldid=17211