An Integral Representation of a Family of Slit Mappings

Adrian W. Cartier, Michael P. Sterner

Department of Biology-Chemistry-Mathematics, University of Montevallo, Montevallo, USA.

DOI: 10.4236/apm.2012.23028                

Abstract

We consider a normalized family F of analytic functions f, whose common domain is the complement of a closed ray in the complex plane. If f(z) is real when z is real and the range of f does not intersect the nonpositive real axis, then f can be reproduced by integrating the biquadratic kernel against a probability measure u(t) . It is shown that while this integral representation does not characterize the family F, it applies to a large class of functions, including a collection of functions which multiply the Hardy space Hp into itself.

Keywords

Herglotz Formula; Integral Representations; Subordination; Slit Mappings; Hardy Spaces; Multipliers; Hadamard Product

Conflicts of Interest

The authors declare no conflicts of interest.

References

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