Bohr’s theorem

(Bohr 1914).  If the power seriesMathworldPlanetmath n=0anzn satisfies

|n=0anzn|< 1 (1)

in the unit disk|z|<1,  then (1) and the inequality

n=0|anzn|< 1 (2)

is true in the disk  |z|<13.  Here, the radius 13 is the best possible.

Proof.  One needs Carathéodory’s inequality which says that if the real partDlmfPlanetmath of a holomorphic functionMathworldPlanetmath


is positive in the unit disk, then

|bn| 2Reb0forn=1, 2,

Choosing now  g(z):=1-eiφf(z)  where φ is any real number and f(z) the sum functionMathworldPlanetmath of the series in the theorem, we get

|an| 2Re(1-eiφa0)= 2(1-a0cosφ),

and especially

|an| 2(1-|a0|),forn=1, 2,

If  f(z)a0,  in the disk  |z|<13  we thus have

n=0|anzn|<|a0|+2(1-|a0|)n=1(13)n= 1.

Take then in particular the function defined by


with  0<c<1.  Its series expansion


shows that


which last form can be seen to become greater than 1 for  |z|>11+2c.  Because c may come from below arbitrarily to 1, one sees that the value 13 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).
Title Bohr’s theorem
Canonical name BohrsTheorem
Date of creation 2015-04-13 12:52:55
Last modified on 2015-04-13 12:52:55
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Theorem
Classification msc 40A30
Classification msc 30B10