View source for Strip method (analytic functions)

A method in the theory of functions of a complex variable that is based on inequalities relating the lengths of curves of a certain special family and the area of the domain occupied by this family. The method is based on Grötzsch' lemmas . One of them is formulated as follows.

Consider a rectangle with sides of lengths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090500/s0905001.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090500/s0905002.png" /> which contains a finite number of non-overlapping simply-connected domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090500/s0905003.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090500/s0905004.png" />, each one having a Jordan boundary that meets the sides of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090500/s0905005.png" /> in segments which do not degenerate into points (the regions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090500/s0905006.png" /> form strips running from one side of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090500/s0905007.png" /> to the other). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090500/s0905008.png" /> is conformally mapped into a rectangle with sides of lengths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090500/s0905009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090500/s09050010.png" /> such that the above segments become the sides of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090500/s09050011.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/s/s090/s090500/s09050012.png" /></td> </tr></table>

with equality attained only if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090500/s09050013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090500/s09050014.png" />, are rectangles with sides of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090500/s09050015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090500/s09050016.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090500/s09050017.png" />.

Another lemma is the [[Grötzsch principle|Grötzsch principle]]. The Grötzsch lemmas are true also for an infinite set of subdomains.

The strip method as a method in the theory of univalent conformal and quasi-conformal mapping was first used by H. Grötzsch , who used the method in a systematic study and solved numerous extremal problems for univalent functions defined in finitely-connected and infinitely-connected domains (see [[#References|[3]]]; for other applications, see [[#References|[2]]]).

The method also forms the basis of the method of the extremal metric (cf. [[Extremal metric, method of the|Extremal metric, method of the]]).

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  H. Grötzsch,   "Über einige Extremalprobleme der konformen Abbildung I"  ''Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Phys. Kl.'' , '''80''' :  6  (1928)  pp. 367–376</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  H. Grötzsch,   "Über die Verzerrung bei schlichten nichtkonformen Abbildungen und über eine damit zusammenhängende Erweiterung des Picardschen Satzes"  ''Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Phys. Kl.'' , '''80''' :  7  (1929)  pp. 503–507</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top">  H. Grötzsch,   "Über die Verzerrung bei schlichter konformer Abbildung mehrfach zusammenhängender schlichter Bereiche"  ''Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Phys. Kl.'' , '''81''' :  1  (1929)  pp. 38–48</TD></TR><TR><TD valign="top">[1d]</TD> <TD valign="top">  H. Grötzsch,   "Über konforme Abbildung unendlichvielfach zusammenhängender schlichter Bereiche mit endlichvielen Häufungsrandkomponenten"  ''Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Phys. Kl.'' , '''81''' :  2  (1929)  pp. 51–87</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Goluzin,   "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Jenkins,   "Univalent functions and conformal mapping" , Springer  (1958)</TD></TR></table>