Integral representations of bounded starlike functions
Frode Rønning
Annales Polonici Mathematici, Tome 62 (1995), p. 289-297 / Harvested from The Polish Digital Mathematics Library
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For α ≥ 0 let ℱα 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 ℱ0. In general, it is only known that bounded starlike functions belong to ℱα for α > 0.

Publié le : 1995-01-01
EUDML-ID : urn:eudml:doc:262370 
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Frode Rønning. Integral representations of bounded starlike functions. Annales Polonici Mathematici, Tome 62 (1995) pp. 289-297. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv60z3p289bwm/

[000] [1] D. A. Brannan and W. E. Kirwan, On some classes of bounded univalent functions, J. London Math. Soc. (2) 1 (1969), 431-443. | Zbl 0177.33403

[001] [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. | Zbl 0227.30013

[002] [3] P. L. Duren, Theory of Hp Spaces, Academic Press, New York, 1970. | Zbl 0215.20203

[003] [4] W. Gröbner und N. Hofreiter, Integraltafel, 2ter Teil, Bestimmte Integrale, Springer, Wien, 1958.

[004] [5] R. A. Hibschweiler and T. H. MacGregor, Closure properties of families of Cauchy-Stieltjes transforms, Proc. Amer. Math. Soc. 105 (1989), 615-621. | Zbl 0669.30028

[005] [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. | Zbl 0703.30015

[006] [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. | Zbl 0823.30007

[007] [8] W. Ma and D. Minda, Uniformly convex functions, Ann. Polon. Math. 57 (1992), 165-175. | Zbl 0760.30004

[008] [9] T. H. MacGregor, Analytic and univalent functions with integral representation involving complex measures, Indiana Univ. Math. J. 36 (1987), 109-130. | Zbl 0644.30009

[009] [10] F. Rønning, A survey on uniformly convex and uniformly starlike functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A, to appear. | Zbl 0879.30004

[010] [11] F. Rønning, On starlike functions associated with parabolic regions, ibid. 45 (14) (1991), 117-122. | Zbl 0769.30011

[011] [12] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), 189-196. | Zbl 0805.30012