Schottky theorem

If a function

$$ \tag{* } w = f( z) = c _ {0} + c _ {1} z + \dots $$

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)