Distortion theorems

under conformal mapping of planar domains

Theorems characterizing the distortion of line elements at a given point of a domain, as well as the distortion of the domain and its subsets, and the distortion of the boundary of the domain under a conformal mapping. Estimates of the modulus of the derivatives of an analytic function at a point of a domain belong first of all to distortion theorems. The statement, for functions in the class of functions

meromorphic and univalent in , that for all , , the inequality

(1)

holds, is a distortion theorem.

Equality at the left-hand side of (1) holds only for the functions

while at the right-hand side equality holds only for the functions

Here and are two arbitrary fixed numbers. The functions map the domain onto the -plane with slit along the interval connecting the points and . The functions map the domain onto the -plane with slit along an arc of the circle with mid-point . Inequality (1) is easily obtained from the Grunsky inequality

which determines the range of values of the functional on the class . On the other hand, inequality (1) is a direct consequence of Goluzin's theorem: If , then for any two points with , , the sharp inequality

(2)

holds, where, moreover, the equality sign is attained for the functions , where is a real constant. Inequality (2) also implies the chord-distortion theorem (cf. [1]). If , then for any two points on the circle the sharp inequality

holds. Equality in this case is only attained for the functions

where is a constant and . Various generalizations of (2) are known. These give the ranges of values of corresponding functionals and are sharpened versions of distortion theorems for or its subclasses (cf., e.g., [1]).

In the class of functions

that are regular and univalent in the disc , the following sharp inequalities are valid for :

(3)

(4)

(5)

The estimates (4) and (5) follow from (3). The inequalities (3)–(5) are called the distortion theorems for . The lower bounds are realized only by the functions

while the upper bounds are realized only by the functions

where . The functions , , known as the Koebe functions, map the disc onto the -plane with slit along the ray , . They are extremal in a number of problems in the theory of univalent functions. Koebe's -theorem holds: The domain that is the image of the disc under a mapping , , always contains the disc , and the point lies on the boundary of this domain only for .

The estimates (3)–(5) are simple consequences of results on the ranges of the functionals

on (cf. [2]).

Let be the class of functions with for . Between functions in and there is the following relation: If , then , and, conversely, if , then . Hence, the range of some functional (or system of functionals) on is determined by the range of the corresponding functional (system of functionals) on , vice versa. E.g., the range of , , on is easily obtained from that of , , on .

For functions that are regular and bounded in a disc, the Schwarz lemma (cf. [1]) and its generalizations, as well as the following boundary-distortion theorem of Löwner are examples of distortion theorems. Löwner's theorem: For a function that is regular in with , in and on an arc of , the length of the image of is not smaller than the length of itself, and equality only holds for the functions , with a real number.

In the class of functions that are univalent in a given multiply-connected domain, the minimum (respectively, maximum) modulus of the derivative at a given point of the domain is attained only for mappings of this domain onto a domain with radial (resp. circular concentric) slits. For unbounded mappings the following theorem holds: Let be a finitely-connected domain in the -plane containing the point at infinity, let be the class of univalent functions in that have in a neighbourhood of the expansion

and let be a point in . Let , , be a function in mapping onto the plane with slits along the arcs of the logarithmic spirals that make an angle with rays emanating from the origin (it is a sufficient to take ; for the logarithmic spiral degenerates into a ray emanating from the origin, while for it degenerates into a circle with centre at the origin). Let

where those branches of the square root are taken that give first coefficients 1 in the Laurent expansions of and in a neighbourhood of . Then the range of on is the disc defined by

where to each boundary point only the functions with suitable , and a constant, correspond. In particular, one has the sharp inequalities

References

[1]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[2]	J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958)
[3]	V.V. Chernikov, "Extremal properties of univalent conformal mappings" , Results of investigation in mathematics and mechanics during 50 years: 1917–1967 , Tomsk (1967) pp. 23–51 (In Russian)
[4]	I.E. Bazilevich, , Mathematics in the USSR during 40 years: 1917–1957 , 1 , Moscow (1959) pp. 444–472 (In Russian)
[5]	P.P. Belinskii, "General properties of quasi-conformal mappings" , Novosibirsk (1974) (In Russian)
[6]	R. Kühnau, "Verzerrungssätze und Koeffizientenbedingungen vom Grunskyschen Typ für quasikonforme Abbildungen" Math. Nachrichten , 48 (1971) pp. 77–105

Comments

Other distortion theorems are, e.g., Landau's theorems (cf. Landau theorems), Bloch's theorem (cf. Bloch constant) and the Pick theorem.

References

[a1]	P.L. Duren, "Univalent functions" , Springer (1983) pp. Chapt. 3
[a2]	C. Pommerenke, "Univalent functions" , Vandenhoeck & Ruprecht (1975)