Blaschke factor

Let be the open unit disc in the complex plane . A holomorphic function

on is called a Blaschke factor if it occurs in a Blaschke product

The defining properties of a Blaschke factor are:

a) a Blaschke factor has precisely one zero in ;

b) a Blaschke factor has norm on the boundary of .

The properties a)–b) may be used to define Blaschke factors on a Dirichlet domain in a Riemann surface as . Here, is the Green function for at and is its (multiple-valued) harmonic conjugate. See [a1] for the planar case.

Thus, in general a Blaschke factor will not be single valued, but it is single valued on simply-connected domains.

Next, for functions of the Nevanlinna class (cf. Boundary properties of analytic functions), the term "Blaschke factor" is used to indicate the Blaschke product that has the same zeros as . For example, on the disc one has the decomposition formula

where is a Blaschke product or the Blaschke factor, is the outer factor, and , are singular inner functions; cf. [a2], Boundary properties of analytic functions; Hardy classes.

Similar decomposition theorems are known for domains in Riemann surfaces, cf. [a3].

References

[a1]	S.D. Fischer, "Function thory on planar domains" , Wiley (1983)
[a2]	J.B. Garnett, "Bounded analytic functions" , Acad. Press (1981)
[a3]	M. Voichick, L. Zalcman, "Inner and outer functions on Riemann Surfaces" Proc. Amer. Math. Soc. , 16 (1965) pp. 1200–1204