Landau's constants
In complex analysis, a branch of mathematics, Landau's constants (Edmund Landau 1929) are certain mathematical constants that describe the behaviour of holomorphic functions defined on the unit disk. Consider the set F of all those holomorphic functions f on the unit disk for which
- [math]\displaystyle{ f'(0) = 1.\, }[/math]
We define Lf to be the radius of the largest disk contained in the image of f, and Bf to be the radius of the largest disk that is the biholomorphic image of a subset of a unit disk.
Landau's constants are then defined as the infimum of Lf or Bf, where f is any holomorphic function or any injective holomorphic function on the unit disk with
- [math]\displaystyle{ f'(0) = 1.\, }[/math]
The three resulting constants are abbreviated L, B, and A (for injective functions), respectively.
The exact values of L, B, and A are not known, but it is known that
- [math]\displaystyle{ 0.4330 + 10^{-14} \lt B \lt 0.472 \,\! }[/math]
- [math]\displaystyle{ {{\sqrt{3}}\over{4}} =0.4330127019....\lt B \le {\left({1\over {\sqrt{1+\sqrt{3}}}} \right) \left( { {\Gamma \left({1\over3} \right) \Gamma \left({11\over12} \right) }\over{ \Gamma \left({1\over4}\right)} } \right) }\lt 0.472 \lt {1 \over 2} \lt L }[/math]
B is the Bloch's constant.
- [math]\displaystyle{ 0.5 \lt L \le {{ \Gamma{{1}\over{3}} \Gamma{{5}\over{6}}}\over{\Gamma{{1}\over{6}}}} = 0.543258965342... \,\! }[/math] (sequence A081760 in the OEIS)
- [math]\displaystyle{ 0.5 \lt A \le 0.7853 }[/math]
See also
- Table of selected mathematical constants
- Bloch's constant
References
- Landau, Edmund (1929), "Über die Blochsche Konstante und zwei verwandte Weltkonstanten", Mathematische Zeitschrift 30 (1): 608–634, doi:10.1007/BF01187791
External links
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