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Schottky theorem

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If a function

is regular and analytic in the disc D = \{ {z } : {| z | < R } \} and does not take certain finite values a _ {1} , a _ {2} in D , then in any disc | z | \leq R _ {1} , 0 < R _ {1} < R , the modulus | f( z) | is bounded by a number M( a _ {1} , a _ {2} , c _ {0} , R _ {1} ) that depends only on a _ {1} , a _ {2} , c _ {0} , R _ {1} ( see [1]). A more complete formulation can be obtained, by combining the generalized Schottky theorem and Landau's theorem, for an arbitrary number q \geq 2 of exceptional values. Suppose that the function (*) does not take some finite values a _ {1} \dots a _ {q} , q \geq 2 . Then for c _ {1} \neq 0 , the radius R is bounded above by a number that depends only on a _ {1} \dots a _ {q} , c _ {0} , c _ {1} ( Landau's theorem). Moreover, in the disc | z | \leq R _ {1} , 0 < R _ {1} < R , the modulus | f( z) | is bounded by a number M( a _ {1} \dots a _ {q} , c _ {0} , R _ {1} ) that depends only on a _ {1} \dots a _ {q} , c _ {0} , R _ {1} ( Schottky's theorem).

From the geometrical point of view Schottky's theorem means that the spherical distance (i.e. the distance on the Riemann sphere) of the image of the disc | z | \leq R _ {1} from the points a _ {1} \dots a _ {q} is no less than a number d( a _ {1} \dots a _ {q} , c _ {0} , R _ {1} ) that depends only on a _ {1} \dots a _ {q} , c _ {0} , R _ {1} . Schottky's theorem is one of the classical results in the theory of functions of a complex variable of the type of distortion theorems.

References

[1] F. Schottky, "Ueber den Picard'schen Satz und die Borel'schen Ungleichungen" Sitzungsber. Preuss. Akad. Wiss. , 2 (1904) pp. 1244–1262
[2] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[3] S. Stoilov, "The theory of functions of a complex variable" , 1–2 , Moscow (1962) (In Russian; translated from Rumanian)

Comments

The theorems of Landau and Schottky are also related to the Picard theorem.

References

[a1] E. Landau, D. Gaier, "Darstellung und Begrundung einiger neuerer Ergebnisse der Funktionentheorie" , Springer, reprint (1986)
[a2] J.B. Conway, "Functions of one complex variable" , Springer (1978)
How to Cite This Entry:
Schottky theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schottky_theorem&oldid=48621
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article