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An Integral Representation of a Family of Slit Mappings
Department of Biology-Chemistry-Mathematics, University of Montevallo, Montevallo, USA
Department of Biology-Chemistry-Mathematics, University of Montevallo, Montevallo, USA
- 1 Department of Biology-Chemistry-Mathematics, University of Montevallo, Montevallo, USA
- 2 Department of Biology-Chemistry-Mathematics, University of Montevallo, Montevallo, USA
Advances in Pure Mathematics·Volume 02 (2012)·Pages 200–202·Published 29 May 2012·DOI10.4236/apm.2012.23028
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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.
KeywordsHerglotz FormulaIntegral RepresentationsSubordinationSlit MappingsHardy SpacesMultipliersHadamard Product
- P. L. Duren, “Univalent Functions,” Springer-Verlag, New York, 1983.
- D. J. Hallenbeck and T. H. MacGregor, “Linear Problems and Convexity Techniques in Geometric Function The- ory,” Pitman Publishing Ltd., London, 1984.
- D. A. Brannan, J. G. Clunie and W. E. Kirwan, “On the Coefficient Problem for Functions of Bounded Boundary Rotation,” Annales Academiae Scientiarum Fennicae. Series AI. Mathematica, Vol. 523, 1972, pp. 403-489.
- T. H. MacGregor and M. P. Sterner, “Hadamard Products with Power Functions and Multipliers of Hardy Spaces,” Journal of Mathematical Analysis and Applications, Vol. 282, No. 1, 2003, pp. 163-176. doi:10.1016/S0022-247X(03)00128-8
- P. L. Duren, “Theory of Hp Spaces,” Academic Press, New York, 1970.