Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Coefficients of symmetric functions of bounded boundary rotation


Author: Wolfram Koepf
Journal: Proc. Amer. Math. Soc. 105 (1989), 324-329
MSC: Primary 30C50
DOI: https://doi.org/10.1090/S0002-9939-1989-0930244-7
MathSciNet review: 930244
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The well-known inclusion relation between functions with bounded boundary rotation and close-to-convex functions of some order is extended to $ m$-fold symmetric functions. This leads solving the corresponding result for close-to-convex functions to the sharp coefficient bounds for $ m$-fold symmetric functions of bounded boundary rotation at most $ k\pi $ when $ k \geq 2m$. Moreover it shows that an $ m$-fold symmetric function of bounded boundary rotation at most $ (2m + 2)\pi $ is close-to-convex and thus univalent.


References [Enhancements On Off] (What's this?)

  • [1] D. Aharonov and S. Friedland, On an inequality connected with the coefficient conjecture for functions of bounded boundary rotation, Ann. Acad. Sci. Fenn. Ser. AI Math. 524 (1973), 1-14. MR 0322155 (48:519)
  • [2] A. Bielecki and Z. Lewandowski, Sur un théorème concernant les fonctions univalentes linéairement accessibles de M. Biernacki, Ann. Polon. Math 12 (1962), 61-63. MR 0147636 (26:5151)
  • [3] D. A. Brannan, On coefficient problems for certain power series, Proceedings of the Symposium on Complex Analysis, Canterbury, 1973, (J. Clunie and W. K. Hayman, eds.), London Math. Soc. Lecture Note Series no. 12, Cambridge Univ. Press, 1974, pp. 17-27. MR 0412411 (54:537)
  • [4] D. A. Brannan, J. G. Clunie and W. E. Kirwan, On the coefficient problem for functions of bounded boundary rotation, Ann. Acad. Sci. Fenn. Ser. AI Math. 523 (1973), 1-18. MR 0338343 (49:3108)
  • [5] L. Brickman, D. J. Hallenbeck, T. H. MacGregor and D. R. Wilken, Convex hulls and extreme points of families of starlike and convex mappings, Trans. Amer. Math. Soc. 185 (1973), 413-428. MR 0338337 (49:3102)
  • [6] P. L. Duren, Univalent functions, Springer-Verlag, New York-Berlin-Heidelberg-Tokyo, 1983. MR 708494 (85j:30034)
  • [7] W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J. 1 (1952), 169-185. MR 0054711 (14:966e)
  • [8] W. Koepf, On the Fekete-Szegö problem for close-to-convex functions, Proc. Amer. Math. Soc. 101 (1987), 89-95. MR 897076 (88i:30015)
  • [9] -, On the Fekete-Szegö problem for close-to-convex functions. II, Arch. Math. 49 (1987), 420-433. MR 915916 (89a:30005)
  • [10] V. Paatero, Über die konforme Abbildung von Gebieten deren Ränder von beschränkter Drehung sind., Ann. Acad. Sci. Fen. Ser. A 33: 9, 1931, 1-78.
  • [11] Ch. Pommerenke, On the coefficients of close-to-convex functions, Michigan Math. J. 9 (1962), 259-269. MR 0147638 (26:5153)
  • [12] -, On close-to-convex analytic functions, Trans. Amer. Math. Soc. 114 (1965), 176-186. MR 0174720 (30:4920)
  • [13] M. S. Robertson, On the theory of univalent functions, Ann. of Math. (2) 37 (1936), 374-408. MR 1503286
  • [14] G. Schober, Univalent functions-selected topics, Lecture Notes in Math., vol. 478, Springer-Verlag, Berlin-Heidelberg-New York, 1975. MR 0507770 (58:22527)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30C50

Retrieve articles in all journals with MSC: 30C50


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0930244-7
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society