inner function

If f:𝔻 is an analytic functionMathworldPlanetmath on the unit disc, we denote by f*(eiθ) the radial limit of f where it exists, that is


A bounded analytic function on the disc will have radial limits almost everywhere (with respect to the Lebesgue measureMathworldPlanetmath on the 𝔻).


A bounded analytic function f is called an inner function if |f*(eiθ)|=1 almost everywhere. If f has no zeros on the unit disc, then f is called a singular inner function.


Every inner function can be written as


where μ is a positive singular measureMathworldPlanetmath on D, B(z) is a Blaschke productMathworldPlanetmath and |α|=1 is a constant.

Note that all the zeros of the function come from the Blaschke product.




where h is a real valued Lebesgue integrableMathworldPlanetmath function on the unit circle and m is the Lebesgue measure. Then f is called an outer function.

The significance of these definitions is that every bounded holomorphic functionMathworldPlanetmath can be written as an inner function times an outer function. See the factorization theorem for H functions (


  • 1 John B. Conway. . Springer-Verlag, New York, New York, 1995.
Title inner function
Canonical name InnerFunction
Date of creation 2013-03-22 15:36:20
Last modified on 2013-03-22 15:36:20
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 6
Author jirka (4157)
Entry type Definition
Classification msc 30H05
Related topic FactorizationTheoremForHinftyFunctions
Defines singular inner function
Defines outer function