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1994-1995 | 60 | 3 | 289-297
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Integral representations of bounded starlike functions

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EN
Abstrakty
EN
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.
Twórcy
  • Trondheim College of Education, Rotvoll Allé, N-7005, Trondheim, Norway
Bibliografia
  • [1] D. A. Brannan and W. E. Kirwan, On some classes of bounded univalent functions, J. London Math. Soc. (2) 1 (1969), 431-443.
  • [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.
  • [3] P. L. Duren, Theory of $H^p$ Spaces, Academic Press, New York, 1970.
  • [4] W. Gröbner und N. Hofreiter, Integraltafel, 2ter Teil, Bestimmte Integrale, Springer, Wien, 1958.
  • [5] R. A. Hibschweiler and T. H. MacGregor, Closure properties of families of Cauchy-Stieltjes transforms, Proc. Amer. Math. Soc. 105 (1989), 615-621.
  • [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.
  • [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.
  • [8] W. Ma and D. Minda, Uniformly convex functions, Ann. Polon. Math. 57 (1992), 165-175.
  • [9] T. H. MacGregor, Analytic and univalent functions with integral representation involving complex measures, Indiana Univ. Math. J. 36 (1987), 109-130.
  • [10] F. Rønning, A survey on uniformly convex and uniformly starlike functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A, to appear.
  • [11] F. Rønning, On starlike functions associated with parabolic regions, ibid. 45 (14) (1991), 117-122.
  • [12] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), 189-196.
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Bibliografia
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bwmeta1.element.bwnjournal-article-apmv60z3p289bwm
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