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''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|Löwner equation]] to solve extremal problems. The method was proposed by K. Löwner [[#References|[1]]]. It is based on the fact that the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l0609601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l0609602.png" />, that are regular and univalent in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l0609603.png" /> and that map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l0609604.png" /> onto domains of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l0609606.png" /> (cf. [[Smirnov domain|Smirnov domain]]), which are obtained from the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l0609607.png" /> by making a slit along a part of a Jordan arc starting from a point on the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l0609608.png" /> and not passing through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l0609609.png" />, is complete (in the topology of uniform convergence of functions inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096010.png" />) in the whole family of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096012.png" />, that are regular and univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096013.png" /> and are such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096015.png" />. Associating the length of the arc that has been removed with a parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096016.png" />, it has been established that a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096018.png" />, that maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096019.png" /> univalently onto a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096020.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096021.png" /> is a solution of the differential equation (see [[Löwner equation|Löwner equation]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096023.png" />, satisfying the initial condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096024.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096026.png" /> is a continuous complex-valued function on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096027.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096028.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096029.png" />. Löwner used this method to obtain sharp estimates of the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096032.png" /> in the expansions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096033.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096034.png" /></td> </tr></table>

in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096035.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096038.png" />, that are regular and univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096039.png" />.

The Löwner method has been used (see [[#References|[3]]]) to obtain fundamental results in the theory of univalent functions (distortion theorems, reciprocal growth theorems, rotation theorems). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096040.png" /> be the subclass of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096041.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096042.png" /> that have in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096043.png" /> the representation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096044.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096045.png" />, as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096046.png" />, is regular and univalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096048.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096051.png" />, and as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096053.png" />, is a solution of the differential equation (*) satisfying the initial condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096054.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096055.png" /> in (*) is any complex-valued function that is piecewise continuous and has modulus 1 on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096056.png" />. To estimate any quantity on the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096057.png" /> it is sufficient to estimate it on the subclass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096058.png" />, since any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096059.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096060.png" /> can be approximated by functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096063.png" />, each of which maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096064.png" /> univalently onto the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096065.png" />-plane with a slit along a Jordan arc starting at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096066.png" /> and not passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096067.png" />, and hence by functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096068.png" />. Under this approximation the quantities to be estimated for the approximating functions converge to the same quantity as for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060960/l06096069.png" />.

Löwner's method has been used in work on the theory of univalent functions (see [[#References|[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 [[#References|[3]]] and [[Variation-parametric method|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 [[#References|[4]]]).

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



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. de Branges,   "A proof of the Bieberbach conjecture"  ''Acta. Math.'' , '''154'''  (1985)  pp. 137–152</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.H. FitzGerald,   C. Pommerenke,   "The de Branges theorem on univalent functions"  ''Trans. Amer. Math. Soc.'' , '''290'''  (1985)  pp. 683–690</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.K. Hayman,   "Multivalent functions" , Cambridge Univ. Press  (1958)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Pommerenke,   "Univalent functions" , Vandenhoeck &amp; Ruprecht  (1975)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P.L. Duren,   "Univalent functions" , Springer  (1983)  pp. 258</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  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</TD></TR></table>