View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l0581501.png" /> be a regular function in an open domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l0581502.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l0581503.png" /> be the number of roots of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l0581504.png" /> located in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l0581505.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l0581506.png" /> be the total length of the curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l0581507.png" /> on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l0581508.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l0581509.png" /> be the area of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l05815010.png" /> and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l05815011.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l05815012.png" /></td> </tr></table>

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

The length-area principle is employed, for example, in the study of properties of regular functions in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l05815013.png" />. In particular, it is employed to prove the following theorem [[#References|[2]]]: If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l05815014.png" /> is regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l05815015.png" /> and has not more than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l05815016.png" /> zeros in it, of which not more than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l05815017.png" /> lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l05815018.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l05815019.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l05815020.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l05815021.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l05815022.png" /> is a constant which depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l05815023.png" />.

The length-area principle and its various generalizations (e.g. the length-volume principle) are also applied in the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l05815024.png" />-dimensional spaces to quasi-conformal mappings, and also to mappings with a bounded [[Dirichlet integral|Dirichlet integral]] [[#References|[4]]]–[[#References|[7]]].

The derivation of the principle involves the use of the [[Bunyakovskii inequality|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|Extremal metric, method of the]]; see, for example, [[#References|[8]]]). This method was used in a less refined form (the strip method, cf. [[Strip method (analytic functions)|Strip method (analytic functions)]]) around the year 1930 to study the properties of mappings of simply- and multiply-connected domains referred to above.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.W. Ahlfors,   "Untersuchungen zur Theorie der konformen Abbildung und der ganzen Funktionen"  ''Acta Soc. Sci. Fennica (A1)'' , '''9'''  (1930)  pp. 1–40</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W.K. Hayman,   "Multivalent functions" , Cambridge Univ. Press  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Nevanilinna,   "Analytic functions" , Springer  (1970)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G.D. Suvorov,   "Families of plane topological mappings" , Novosibirsk  (1965)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.A. Krein,   ''Mat. Sb.'' , '''9''' :  3  (1941)  pp. 713–719</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  I.S. Ovchinnikov,   "An inequality of the type of the length-area principle for mappings that leave certain integral functionals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058150/l05815025.png" />-dimensional space bounded" , ''Metric questions in the theory of functions and mappings'' , '''3''' , Moscow  (1971)  pp. 98–115  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J. Lelong-Ferrand,   "Représentation conforme et transformations à intégrale de Dirichlet bornée" , Gauthier-Villars  (1955)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  J.A. Jenkins,   "Univalent functions and conformal mapping" , Springer  (1958)</TD></TR></table>



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