View source for Symmetrization method

{{TEX|done}}
''(in function theory)''

One of the methods for solving extremal problems in the geometric theory of functions. Underlying the method is the notion of [[Symmetrization|symmetrization]] of closed and open sets of the $n$-dimensional Euclidean space. Symmetrization methods in function theory were first applied in the study of the properties of the transfinite diameter (see [[#References|[1]]]), somewhat later to the solution of the Carleman–Milloux problem (see [[#References|[2]]]) and were then used quite widely (see [[#References|[3]]]–[[#References|[6]]], [[#References|[9]]]).

The use of symmetrization methods in function theory is based on the monotone nature of the change of the capacity and interior radius of a domain under various forms of symmetrization. The possibility of applying symmetrization methods in solving extremal problems in the geometric theory of functions results from a definite symmetry of the extremal transformations. Based on the property of non-decrease of the interior radius of a domain under symmetrization relative to a line or ray and using the theorem on the change of the interior radius of a domain under transformation by a regular function, the following symmetrization principle was obtained (see [[#References|[4]]]): If the function $w=f(z)$, $f(0)=w_0$, $f'(0)=a_1$, is regular in the disc $E$: $|z|<1$, if $E_f$ is the set of values of $w=f(z)$ in $E$, if $E_f^*$ is the result of symmetrization of $E_f$ relative to a ray or line passing through $w=w_0$, and if $r(E_f^*,w_0)$ is the interior radius of $E_f^*$ relative to the point $w_0$, then

$$r(E_f^*,w_0)\geq|a_1|.\label{*}\tag{*}$$

Equality holds in \eqref{*} if and only if $w=f(z)$ is univalent in $E$ and $E_f^*$ coincides with $E_f$ (under Steiner symmetrization) or can be obtained from $E_f$ as a result of rotation around $w_0$ (under Pólya symmetrization). A similar result holds for other forms of symmetrization for which the property of non-decrease of the interior radius holds. Additional research is usually necessary to clarify the condition for equality in \eqref{*}.

There are generalizations of the symmetrization principle to the case of an annulus and domains of arbitrary connectivity (see [[#References|[6]]]). It is fruitful to combine symmetrization methods and other methods for solving extremal problems in the geometric theory of functions (cf. [[Extremal metric, method of the|Extremal metric, method of the]]; [[Quadratic differential|Quadratic differential]], etc.). A number of covering and distortion theorems for various classes of functions regular in a given domain (univalent, univalent in the mean, weakly $p$-valent in the disc or in an annulus, etc., see [[#References|[4]]]–[[#References|[6]]]) have been obtained by this route.

Symmetrization methods have also been applied to the study of properties of spaces of quasi-conformal mappings. This situation is particularly important in view of the limited methods available for studying such mappings. Symmetrization methods allow one to find, among the doubly-connected spatial domains with fixed geometric properties, domains with largest conformal modulus. The determination of such a domain, in its turn, allows one to establish an extremal property of a quasi-conformal mapping. In particular, by the use of a symmetrization method, certain distortion theorems have been established for quasi-conformal mappings in three-dimensional Euclidean space (see [[#References|[7]]], [[#References|[8]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Faber,   "Ueber Potentialtheorie und konforme Abbildung"  ''Sitzungsber. Bayer. Akad. Wiss. Math.-Naturwiss. Kl.'' , '''20'''  (1920)  pp. 49–64</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Beurling,   "Etudes sur un problème de majoration"  L. Carleson (ed.)  et al. (ed.) , ''Collected Works'' , Birkhäuser  (1989)  pp. 1–108  (Thèse)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Pólya,   G. Szegö,   "Isoperimetric inequalities in mathematical physics" , Princeton Univ. Press  (1951)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W.K. Hayman,   "Multivalent functions" , Cambridge Univ. Press  (1958)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.A. Jenkins,   "Univalent functions and conformal mappings" , Springer  (1958)</TD></TR><TR><TD valign="top">[6]</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">[7]</TD> <TD valign="top">  B.V. Shabat,   "On the theory of quasiconformal mappings in space"  ''Soviet Math. Dokl.'' , '''1''' :  3  (1960)  pp. 730–733  ''Dokl. Akad. Nauk SSSR'' , '''132''' :  5  (1960)  pp. 1045–1048</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  F.W. Gehring,   "Symmetrization of rings in space"  ''Trans. Amer. Math. Soc.'' , '''101''' :  3  (1961)  pp. 499–519</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A. Baernstein,   "Integral means, univalent functions and circular symmetrization"  ''Acta Math.'' , '''133'''  (1974)  pp. 139–169</TD></TR></table>