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Two Results Concerning the Zeros of Functions with Finite Dirichlet Integral

Published online by Cambridge University Press:  20 November 2018

James G. Caughran
James G. Caughran*
Affiliation:
University of Kentucky, Lexington, Kentucky
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A function f, analytic in the unit disk, is said to have finite Dirichlet integral if

1

Geometrically, this is equivalent to f mapping the disk onto a Riemann surface of finite area. The class of Dirichlet integrable functions will be denoted by . The condition above can be restated in terms of Taylor coefficients; if f(z) = Σanzn, then if and only if Σn|an|2 < ∞. Thus, is contained in the Hardy class H2.

In particular, every such function has boundary values

almost everywhere and log |f(eiθ)| ∊ L1(dθ).

The zeros zn of a function must satisfy the Blaschke condition

and f(s) = B(z)F(z), where F(z) has no zeros and

is the Blaschke product with zeros zn; see (5).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 312 - 316
Copyright
Copyright © Canadian Mathematical Society 1969

References

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5. Hoffman, Kenneth, Banach spaces of analytic functions (Prentice-Hall, Englewood Cliffs, N.J., 1962).Google Scholar
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