Area principle

The area of the complement to the image of a domain under a mapping by a function regular in it is non-negative. The area principle was first used in 1914 by T.H. Gronwall [1], who in this way proved the so-called exterior area theorem for functions of class — functions

that are regular and univalent in the annulus (cf. Univalent function). The area of the complement of the image of under a mapping can be determined by the formula

and consequently

(1)

By means of (1) the first results were obtained for functions of the classes and , where is the class of functions that are regular and univalent in the disc (cf. Bieberbach conjecture; Distortion theorems). A more general area theorem has been proved [2]. G.M. Goluzin [3] extended the area theorem to -valent functions in the disc (cf. Multivalent function).

The following area theorem has been proved [4]: Let , , and let be a regular function in . If

then

(2)

and equality holds only if the area of is zero.

By the area theorem for a certain class of univalent functions , , with a domain, one usually understands any inequality having the property that equality holds if and only if the area of the complement of is zero, and the same applies for a class of systems of univalent functions where is a domain and is the complement of . Usually, such a theorem is proved by means of the area principle. That is, one considers any regular function , or more generally, one having a regular derivative, on and calculates the area of the image of under the mapping of the function . Therefore, (2) is a certain extremely general area theorem in the class .

Let and let

One chooses a suitable function, regular in , to be able to write (2) as

(3)

where the are any numbers not simultaneously equal to zero and such that

More general area theorems in have also been obtained [5].

Area theorems have been proved for the following: the class of systems of functions that map the disc conformally and univalently onto domains without pairwise common points, i.e. onto non-adjacent domains [6]; the class (, , is the class of functions that are regular and univalent in and such that , ), [7]; and non-overlapping multiple-connected domains (see [6] and also [8], [9]). All the area theorems for multiple-connected domains can be proved by the method of contour integration (cf. Contour integration, method of).

By the area method one understands methods of solving various problems in the theory of univalent functions by the use of area theorems.

For instance, from (3) one may obtain by means of the Cauchy inequality that

(4)

where and are such that the series on the right-hand side converge. If in (4), for example, , , , , one obtains the chord-distortion theorem

Area theorems, for example in the class , give necessary and sufficient conditions for a system of meromorphic functions to belong to the class (see [6], p. 179).

References

[1]	T.H. Gronwall, "Some remarks on conformal representation" Ann. of Math. Ser. 2 , 16 (1914 - 1915) pp. 72–76
[2]	H. Prawitz, "Ueber Mittelwerte analytischer Funktionen" Arkiv. Mat. Astron., Fysik , 20A : 6 (1927) pp. 1–12
[3]	G.M. Goluzin, "On -valued functions" Math. Sb. , 8 : 2 (1940) pp. 277–284 (In Russian)
[4]	N.A. Lebedev, I.M. Milin, "On the coefficients of certain classes of analytic functions" Mat. Sb. , 28 : 2 (1951) pp. 359–400 (In Russian)
[5]	Z. Nehari, "Inequalities for the coefficients of univalent functions" Arch. Rational Mech. and Anal. , 34 : 4 (1969) pp. 301–330
[6]	N.A. Lebedev, "The area principle in the theory of univalent functions" , Moscow (1975) (In Russian)
[7]	I.M. Milin, "Univalent functions and orthonormal systems" , Transl. Math. Monogr. , 49 , Amer. Math. Soc. (1977) (Translated from Russian)
[8]	Yu.E. Alenitsyn, "Area theorems for functions that are analytic in a finitely-connected domain" Izv. Akad. Nauk SSSR Ser. Mat. , 37 : 5 (1973) pp. 1132–1154 (In Russian)
[9]	V.Ya. Gultyanskii, V.A. Shchepetov, "A general area theorem for a certain class of q-quasi-conformal mappings" Dokl. Akad. Nauk SSSR , 218 : 3 (1974) pp. 509–512 (In Russian)
[10]	H. Grunsky, "Koeffizientenbedingungen für schlichtabbildender meromorphe Funktionen" Math. Z. , 45 : 1 (1939) pp. 29–61

Comments

The inequalities (3) and (4) are a form of the Grunsky inequalities.

References

[a1]	P.L. Duren, "Univalent functions" , Springer (1983) pp. Chapt. 10