Bloch constant
From Encyclopedia of Mathematics
An absolute constant, the existence of which is established by Bloch's theorem. Let be the class of all holomorphic functions
in the disc
such that
. The Riemann surface of the function
contains on one of its sheets a largest open disc of radius
. It was shown by A. Bloch [1] that
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The most precise known estimate is [2]. It follows from Bloch's theorem that the Riemann surface of an entire function contains single-sheeted discs of arbitrary radius; this is equivalent to the Picard theorem.
References
[1] | A. Bloch, "Les théorèmes de M. Valiron sur les fonctions entières et la théorie de l'uniformisation" Ann. Fac. Sci. Univ. Toulouse (3) , 17 (1925) pp. 1–22 |
[2] | L.V. Ahlfors, H. Grunsky, "Ueber die Blochsche Konstante" Math. Z. , 42 (1937) pp. 671–673 |
[3] | G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian) |
Comments
For the connection between the theorems of Bloch and Picard, see e.g. [a1].
References
[a1] | M. Heins, "Selected topics in the classical theory of functions of a complex variable" , Holt, Rinehart & Winston (1962) |
How to Cite This Entry:
Bloch constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bloch_constant&oldid=39739
Bloch constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bloch_constant&oldid=39739
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article