TITLE:
An Integral Representation of a Family of Slit Mappings
AUTHORS:
Adrian W. Cartier, Michael P. Sterner
KEYWORDS:
Herglotz Formula; Integral Representations; Subordination; Slit Mappings; Hardy Spaces; Multipliers; Hadamard Product
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.2 No.3,
May
29,
2012
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.