Theorem (Bohr 1914). If the power series satisfies
in the unit disk , then (1) and even the inequality
is true in the disk . Here, the radius is the best possible.
Proof. One needs Carathéodory’s inequality which says that if the real part of a holomorphic function
is positive in the unit disk, then
Choosing now where is any real number and the sum function of the series in the theorem, we get
If , in the disk we thus have
Take then in particular the function defined by
with . Its series expansion
which last form can be seen to become greater than 1 for . Because may come from below arbitrarily near to 1, one sees that the value in the theorem cannot be increased.
- 1 Harald Bohr: “A theorem concerning power series”. – Proc. London Math. Soc. 13 (1914).
- 2 Harold P. Boas: “Majorant series”. – J. Korean Math. Soc. 37 (2000).