# Symmetrization method

*(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 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 [1]), somewhat later to the solution of the Carleman–Milloux problem (see [2]) and were then used quite widely (see [3]–[6], [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 [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 [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; 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 [4]–[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 [7], [8]).

#### References

[1] | G. Faber, "Ueber Potentialtheorie und konforme Abbildung" Sitzungsber. Bayer. Akad. Wiss. Math.-Naturwiss. Kl. , 20 (1920) pp. 49–64 |

[2] | A. Beurling, "Etudes sur un problème de majoration" L. Carleson (ed.) et al. (ed.) , Collected Works , Birkhäuser (1989) pp. 1–108 (Thèse) |

[3] | G. Pólya, G. Szegö, "Isoperimetric inequalities in mathematical physics" , Princeton Univ. Press (1951) |

[4] | W.K. Hayman, "Multivalent functions" , Cambridge Univ. Press (1958) |

[5] | J.A. Jenkins, "Univalent functions and conformal mappings" , Springer (1958) |

[6] | G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian) |

[7] | 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 |

[8] | F.W. Gehring, "Symmetrization of rings in space" Trans. Amer. Math. Soc. , 101 : 3 (1961) pp. 499–519 |

[9] | A. Baernstein, "Integral means, univalent functions and circular symmetrization" Acta Math. , 133 (1974) pp. 139–169 |

**How to Cite This Entry:**

Symmetrization method.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Symmetrization_method&oldid=44734