Coefficients of symmetric functions of bounded boundary rotation
HTML articles powered by AMS MathViewer
- by Wolfram Koepf
- Proc. Amer. Math. Soc. 105 (1989), 324-329
- DOI: https://doi.org/10.1090/S0002-9939-1989-0930244-7
- PDF | Request permission
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
- D. Aharonov and S. Friedland, On an inequality connected with the coefficient conjecture for functions of bounded boundary rotation, Ann. Acad. Sci. Fenn. Ser. A. I. 524 (1972), 14. MR 322155
- A. Bielecki and Z. Lewandowski, Sur un théorème concernant les fonctions univalentes linéairement accessibles de M. Biernacki, Ann. Polon. Math. 12 (1962), 61–63 (French). MR 147636, DOI 10.4064/ap-12-1-61-63
- D. A. Brannan, On coefficient problems for certain power series, Proceedings of the Symposium on Complex Analysis (Univ. Kent, Canterbury, 1973) London Math. Soc. Lecture Note Ser., No. 12, Cambridge Univ. Press, London, 1974, pp. 17–27. MR 0412411
- 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. A. I. 523 (1973), 18. MR 338343
- 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 (1974). MR 338337, DOI 10.1090/S0002-9947-1973-0338337-5
- Peter L. Duren, Univalent functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
- Wilfred Kaplan, Close-to-convex schlicht functions, Michigan Math. J. 1 (1952), 169–185 (1953). MR 54711
- Wolfram Koepf, On the Fekete-Szegő problem for close-to-convex functions, Proc. Amer. Math. Soc. 101 (1987), no. 1, 89–95. MR 897076, DOI 10.1090/S0002-9939-1987-0897076-8
- Wolfram Koepf, On the Fekete-Szegő problem for close-to-convex functions. II, Arch. Math. (Basel) 49 (1987), no. 5, 420–433. MR 915916, DOI 10.1007/BF01194100 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.
- Ch. Pommerenke, On the coefficients of close-to-convex functions, Michigan Math. J. 9 (1962), 259–269. MR 147638
- Ch. Pommerenke, On close-to-convex analytic functions, Trans. Amer. Math. Soc. 114 (1965), 176–186. MR 174720, DOI 10.1090/S0002-9947-1965-0174720-4
- Malcolm I. S. Robertson, On the theory of univalent functions, Ann. of Math. (2) 37 (1936), no. 2, 374–408. MR 1503286, DOI 10.2307/1968451
- Glenn Schober, Univalent functions—selected topics, Lecture Notes in Mathematics, Vol. 478, Springer-Verlag, Berlin-New York, 1975. MR 0507770
Bibliographic Information
- © Copyright 1989 American Mathematical Society
- 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