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 $ A $ and $ B $ which contains a finite number of non-overlapping simply-connected domains $ S _ {k} $, $ k = 1 \dots n $, each one having a Jordan boundary that meets the sides of length $ A $ in segments which do not degenerate into points (the regions $ S _ {k} $ form strips running from one side of length $ A $ to the other). If $ S _ {k} $ is conformally mapped into a rectangle with sides of lengths $ a _ {k} $ and $ b _ {k} $ such that the above segments become the sides of length $ a _ {k} $, then

$$ \sum _ { k= 1} ^ { n } \frac{a _ {k} }{b _ {k} } \leq \frac{A}{B} , $$

with equality attained only if the $ S _ {k} $, $ k = 1 \dots n $, are rectangles with sides of length $ a _ {k} ^ \prime $ and $ B $ with $ \sum _ {k=1} ^ {n} a _ {k} ^ \prime = A $.

Another lemma is the 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 [3]; for other applications, see [2]).

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

References

[1a]	H. Grötzsch, "Über einige Extremalprobleme der konformen Abbildung I" Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Phys. Kl. , 80 : 6 (1928) pp. 367–376
[1b]	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
[1c]	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
[1d]	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
[2]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[3]	J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958)