Difference between revisions of "Strip method (analytic functions)"
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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. | 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 | + | 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 | + | 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|Grötzsch principle]]. The Grötzsch lemmas are true also for an infinite set of subdomains. | Another lemma is the [[Grötzsch principle|Grötzsch principle]]. The Grötzsch lemmas are true also for an infinite set of subdomains. | ||
Revision as of 08:24, 6 June 2020
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) |
Strip method (analytic functions). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strip_method_(analytic_functions)&oldid=48873