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An Integral Representation of a Family of Slit Mappings

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Author(s): Adrian W. Cartier | Michael P. Sterner

Journal: Advances in Pure Mathematics
ISSN 2160-0368

Volume: 02;
Issue: 03;
Start page: 200;
Date: 2012;
Original page

Keywords: Herglotz Formula | Integral Representations | Subordination | Slit Mappings | Hardy Spaces | Multipliers | Hadamard Product

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.
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