Löwner method

Löwner's method of parametric representation of univalent functions, Löwner's parametric method

A method in the theory of univalent functions that consists in using the Löwner equation to solve extremal problems. The method was proposed by K. Löwner [1]. It is based on the fact that the set of functions $ f ( z) $, $ f ( 0) = 0 $, that are regular and univalent in the disc $ E = \{ {z } : {| z | < 1 } \} $ and that map $ E $ onto domains of type $ ( s) $( cf. Smirnov domain), which are obtained from the disc $ | w | < 1 $ by making a slit along a part of a Jordan arc starting from a point on the circle $ | w | = 1 $ and not passing through the point $ w = 0 $, is complete (in the topology of uniform convergence of functions inside $ E $) in the whole family of functions $ f ( z) $, $ f ( 0) = 0 $, that are regular and univalent in $ E $ and are such that $ | f ( z) | < 1 $ in $ E $. Associating the length of the arc that has been removed with a parameter $ t $, it has been established that a function $ w = f ( z) $, $ f ( 0) = 0 $, that maps $ E $ univalently onto a domain $ D $ of type $ ( s) $ is a solution of the differential equation (see Löwner equation)

$$ \tag{* } \frac{\partial f ( z , t ) }{\partial t } = - f ( z , t ) \frac{1 + k ( t) f ( z , t ) }{1 - k ( t) f ( z , t ) } , $$

$ f ( z , t _ {0} ) = f ( z) $, satisfying the initial condition $ f ( z , 0 ) = z $. Here $ t \in [ 0 , t _ {0} ] $ and $ k ( t) $ is a continuous complex-valued function on the interval $ [ 0 , t _ {0} ] $ corresponding to $ D $ with $ | k ( t) | = 1 $. Löwner used this method to obtain sharp estimates of the coefficients $ c _ {3} $ and $ b _ {n} $, $ n = 2 , 3 \dots $ in the expansions

$$ w = f ( z) = z + \sum _ { n= } 2 ^ \infty c _ {n} z ^ {n} $$

and

$$ z = f ^ { - 1 } ( w) = w + \sum _ { n= } 2 ^ \infty b _ {n} w ^ {n} $$

in the class $ S $ of functions $ w = f ( z) $, $ f ( 0) = 0 $, $ f ^ { \prime } ( 0) = 1 $, that are regular and univalent in $ E $.

The Löwner method has been used (see [3]) to obtain fundamental results in the theory of univalent functions (distortion theorems, reciprocal growth theorems, rotation theorems). Let $ S ^ { \prime } $ be the subclass of functions $ f ( z) $ in $ S $ that have in $ E $ the representation

$$ f ( z) = \lim\limits _ {t \rightarrow \infty } \ e ^ {t} f ( z , t ) , $$

where $ f ( z , t ) $, as a function of $ z $, is regular and univalent in $ E $, $ | f ( z , t) | < 1 $ in $ E $, $ f ( 0 ,t ) = 0 $, $ f _ {z} ^ { \prime } ( 0 , t ) > 0 $, and as a function of $ t $, $ 0 < t < \infty $, is a solution of the differential equation (*) satisfying the initial condition $ f ( z , 0 ) = z $; $ k ( t) $ in (*) is any complex-valued function that is piecewise continuous and has modulus 1 on the interval $ [ 0 , \infty ) $. To estimate any quantity on the class $ S $ it is sufficient to estimate it on the subclass $ S ^ { \prime } $, since any function $ f ( z) $ of class $ S $ can be approximated by functions $ f _ {n} ( z) $, $ f _ {n} ( 0) = 0 $, $ f _ {n} ^ { \prime } ( 0) > 0 $, each of which maps $ E $ univalently onto the $ w $- plane with a slit along a Jordan arc starting at $ \infty $ and not passing through $ w = 0 $, and hence by functions $ f _ {n} ( z) / f _ {n} ^ { \prime } ( 0) \in S ^ { \prime } $. Under this approximation the quantities to be estimated for the approximating functions converge to the same quantity as for the function $ f ( z) $.

Löwner's method has been used in work on the theory of univalent functions (see [3]); it often leads to success in obtaining explicit estimates, but as a rule it does not ensure the classification of all extremal functions and does not give complete information about their uniqueness. For a complete solution of extremal problems Löwner's method is usually combined with a variational method (see [3] and Variation-parametric method). Löwner's method has been extended to doubly-connected domains. A generalized equation of the type of Löwner's equation has been obtained for multiply-connected domains and for automorphic functions (see [4]).

References

[1]	K. Löwner, "Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I" Math. Ann. , 89 (1923) pp. 103–121
[2]	E. Peschl, "Zur Theorie der schlichten Funktionen" J. Reine Angew. Math. , 176 (1936) pp. 61–94
[3]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[4]	I.A. Aleksandrov, "Parametric extensions in the theory of univalent functions" , Moscow (1976) (In Russian)

Comments

The Löwner equation has been used to solve the Bieberbach conjecture, [a1]; cf. [a2]. Further references on the method include [a3]–[a6].

References

[a1]	L. de Branges, "A proof of the Bieberbach conjecture" Acta. Math. , 154 (1985) pp. 137–152
[a2]	C.H. FitzGerald, C. Pommerenke, "The de Branges theorem on univalent functions" Trans. Amer. Math. Soc. , 290 (1985) pp. 683–690
[a3]	W.K. Hayman, "Multivalent functions" , Cambridge Univ. Press (1958)
[a4]	C. Pommerenke, "Univalent functions" , Vandenhoeck & Ruprecht (1975)
[a5]	P.L. Duren, "Univalent functions" , Springer (1983) pp. 258
[a6]	D.A. Brannan, "The Löwner differential equation" D.A. Brannan (ed.) J.G. Clunie (ed.) , Aspects of Contemporary Complex Analysis , Acad. Press (1980) pp. 79–95