An Integral Representation of a Family of Slit Mappings

Affiliation(s)

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

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

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.

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.

Cite this paper

A. Cartier and M. Sterner, "An Integral Representation of a Family of Slit Mappings,"*Advances in Pure Mathematics*, Vol. 2 No. 3, 2012, pp. 200-202. doi: 10.4236/apm.2012.23028.

A. Cartier and M. Sterner, "An Integral Representation of a Family of Slit Mappings,"

References

[1] P. L. Duren, “Univalent Functions,” Springer-Verlag, New York, 1983.

[2] D. J. Hallenbeck and T. H. MacGregor, “Linear Problems and Convexity Techniques in Geometric Function The- ory,” Pitman Publishing Ltd., London, 1984.

[3] 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.

[4] 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

[5] P. L. Duren, “Theory of Hp Spaces,” Academic Press, New York, 1970.

[1] P. L. Duren, “Univalent Functions,” Springer-Verlag, New York, 1983.

[2] D. J. Hallenbeck and T. H. MacGregor, “Linear Problems and Convexity Techniques in Geometric Function The- ory,” Pitman Publishing Ltd., London, 1984.

[3] 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.

[4] 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

[5] P. L. Duren, “Theory of Hp Spaces,” Academic Press, New York, 1970.