Length-and-area principle

length-area principle

A principle expressing the connection between the lengths of curves belonging to some specific family and the area covered by this family.

Let $w=f(z)$ be a regular function in an open domain $G$. Let $n(w)$ be the number of roots of the equation $f(z)=w$ located in $G$; let $l(\rho)$ be the total length of the curves in $G$ on which $|f(z)|=\rho$; let $A$ be the area of $G$ and let

$$p(\rho)=\frac{1}{2\pi}\int\limits_0^{2\pi}n(\rho e^{i\theta})d\theta,\quad\rho>0.$$

Then the length-area principle is given by the inequality [2]:

$$\int\limits_0^\infty\frac{l(\rho)^2d\rho}{\rho p(\rho)}\leq2\pi A.$$

The principle has found extensive use in the theory of functions of a complex variable [1]–[4].

The length-area principle is employed, for example, in the study of properties of regular functions in the disc $|z|<1$. In particular, it is employed to prove the following theorem [2]: If the function $w=f(z)=a_0+a_1z+\dotsb$ is regular in $|z|<1$ and has not more than $q$ zeros in it, of which not more than $h$ lie in $|z|<1/2$, and $\mu_q=\max_{0\leq\nu\leq q}|a_\nu|$, then

$$\int\limits_{R_1}^{R_2}\frac{d\rho}{\rho p(\rho)}<2\ln\frac{1}{1-r}+A(q),$$

where

$$R_1=(h+2)2^{h-1}\mu_h,\quad R_2=\max_{|z|=r}|f(z)|,\quad0<r<1,$$

and $A_q$ is a constant which depends on $q$.

The length-area principle and its various generalizations (e.g. the length-volume principle) are also applied in the case of $n$-dimensional spaces to quasi-conformal mappings, and also to mappings with a bounded Dirichlet integral [4]–[7].

The derivation of the principle involves the use of the Bunyakovskii inequality. Subsequent study of the connection between the lengths of curves and the regions they cover led to an important method for studying univalent conformal and quasi-conformal mappings — the method of the extremal metric (cf. Extremal metric, method of the; see, for example, [8]). This method was used in a less refined form (the strip method, cf. Strip method (analytic functions)) around the year 1930 to study the properties of mappings of simply- and multiply-connected domains referred to above.

Comments

Bunyakovskii's inequality is usually called the (Cauchy–) Schwarz (–Bunyakovskii) inequality in the English literature.

References

[1]	L.W. Ahlfors, "Untersuchungen zur Theorie der konformen Abbildung und der ganzen Funktionen" Acta Soc. Sci. Fennica (A1) , 9 (1930) pp. 1–40
[2]	W.K. Hayman, "Multivalent functions" , Cambridge Univ. Press (1958)
[3]	R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)
[4]	G.D. Suvorov, "Families of plane topological mappings" , Novosibirsk (1965) (In Russian)
[5]	M.A. Krein, Mat. Sb. , 9 : 3 (1941) pp. 713–719
[6]	I.S. Ovchinnikov, "An inequality of the type of the length-area principle for mappings that leave certain integral functionals in $n$-dimensional space bounded" , Metric questions in the theory of functions and mappings , 3 , Moscow (1971) pp. 98–115 (In Russian)
[7]	J. Lelong-Ferrand, "Représentation conforme et transformations à intégrale de Dirichlet bornée" , Gauthier-Villars (1955)
[8]	J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958)