Integral representations of bounded starlike functions

Frode Rønning

Annales Polonici Mathematici (1995)

  • Volume: 60, Issue: 3, page 289-297
  • ISSN: 0066-2216

Abstract

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For α ≥ 0 let ℱ α denote the class of functions defined for |z| < 1 by integrating 1 / ( 1 - x z ) α if α > 0, and log(1/(1-xz)) if α = 0, against a complex measure on |x| = 1. We study families of starlike functions where zf’(z)/f(z) ranges over a parabola with given focus and vertex. We prove a number of properties of these functions, among others that they are bounded and that they belong to ℱ 0 . In general, it is only known that bounded starlike functions belong to ℱ α for α > 0.

How to cite

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Frode Rønning. "Integral representations of bounded starlike functions." Annales Polonici Mathematici 60.3 (1995): 289-297. <http://eudml.org/doc/262370>.

@article{FrodeRønning1995,
abstract = {For α ≥ 0 let $ℱ_α$ denote the class of functions defined for |z| < 1 by integrating $1/(1-xz)^α$ if α > 0, and log(1/(1-xz)) if α = 0, against a complex measure on |x| = 1. We study families of starlike functions where zf’(z)/f(z) ranges over a parabola with given focus and vertex. We prove a number of properties of these functions, among others that they are bounded and that they belong to $ℱ_0$. In general, it is only known that bounded starlike functions belong to $ℱ_α$ for α > 0.},
author = {Frode Rønning},
journal = {Annales Polonici Mathematici},
keywords = {Cauchy-Stieltjes integrals; starlike functions; bounded starlike functions},
language = {eng},
number = {3},
pages = {289-297},
title = {Integral representations of bounded starlike functions},
url = {http://eudml.org/doc/262370},
volume = {60},
year = {1995},
}

TY - JOUR
AU - Frode Rønning
TI - Integral representations of bounded starlike functions
JO - Annales Polonici Mathematici
PY - 1995
VL - 60
IS - 3
SP - 289
EP - 297
AB - For α ≥ 0 let $ℱ_α$ denote the class of functions defined for |z| < 1 by integrating $1/(1-xz)^α$ if α > 0, and log(1/(1-xz)) if α = 0, against a complex measure on |x| = 1. We study families of starlike functions where zf’(z)/f(z) ranges over a parabola with given focus and vertex. We prove a number of properties of these functions, among others that they are bounded and that they belong to $ℱ_0$. In general, it is only known that bounded starlike functions belong to $ℱ_α$ for α > 0.
LA - eng
KW - Cauchy-Stieltjes integrals; starlike functions; bounded starlike functions
UR - http://eudml.org/doc/262370
ER -

References

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  1. [1] D. A. Brannan and W. E. Kirwan, On some classes of bounded univalent functions, J. London Math. Soc. (2) 1 (1969), 431-443. Zbl0177.33403
  2. [2] L. Brickman, T. H. MacGregor and D. R. Wilken, Convex hulls of some classical families of univalent functions, Trans. Amer. Math. Soc. 156 (1971), 91-107. Zbl0227.30013
  3. [3] P. L. Duren, Theory of H p Spaces, Academic Press, New York, 1970. Zbl0215.20203
  4. [4] W. Gröbner und N. Hofreiter, Integraltafel, 2ter Teil, Bestimmte Integrale, Springer, Wien, 1958. 
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  6. [6] R. A. Hibschweiler and T. H. MacGregor, Univalent functions with restricted growth and Cauchy-Stieltjes integrals, Complex Variables Theory Appl. 15 (1990), 53-63. Zbl0703.30015
  7. [7] W. Ma and D. Minda, A unified treatment of some special classes of univalent functions, in: Proc. Internat. Conf. of Complex Analysis, to appear. Zbl0823.30007
  8. [8] W. Ma and D. Minda, Uniformly convex functions, Ann. Polon. Math. 57 (1992), 165-175. Zbl0760.30004
  9. [9] T. H. MacGregor, Analytic and univalent functions with integral representation involving complex measures, Indiana Univ. Math. J. 36 (1987), 109-130. Zbl0644.30009
  10. [10] F. Rønning, A survey on uniformly convex and uniformly starlike functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A, to appear. Zbl0879.30004
  11. [11] F. Rønning, On starlike functions associated with parabolic regions, ibid. 45 (14) (1991), 117-122. Zbl0769.30011
  12. [12] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), 189-196. Zbl0805.30012

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