Bloch’s theorem can be stated in the following way:
Let be the set of all functions holomorphic on a region containing the closure of the disk and satisfying and . For each let be the supremum of all numbers such that there is a disk on which is injective and contains a disk of radius . Let be the infimum of all , for . Then .
The constant is usually referred to as Bloch’s constant. Nowadays, better bounds are known and, in fact, it has been conjectured that has the following tantalizing form
where is the gamma function.
- 1 John B. Conway, Functions of One Complex Variable I, Second Edition, 1978, Springer-Verlag, New York.
|Date of creation||2013-03-22 15:58:04|
|Last modified on||2013-03-22 15:58:04|
|Last modified by||alozano (2414)|