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)

 
 

 

On the coefficients of Bazilevič functions


Authors: F. R. Keogh and Sanford S. Miller
Journal: Proc. Amer. Math. Soc. 30 (1971), 492-496
MSC: Primary 30.43
DOI: https://doi.org/10.1090/S0002-9939-1971-0283191-7
MathSciNet review: 0283191
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ B(\alpha )$ denote the class of normalized $ (f(0) = 0,f'(0) = 1)$, Bazilevič functions of type $ \alpha $ defined in $ \Delta :\vert z\vert < 1,{\text{i.e.}}f(z) = {[\alpha \smallint _0^zP(\zeta )g{(\zeta )^\alpha }{\zeta ^{ - 1}}d\zeta ]^{1/\alpha }}$ where $ g(\zeta )$ is starlike in $ \Delta ,P(\zeta )$ is regular with $ \Re \,P(\zeta ) > 0$ in $ \Delta $ and $ \alpha > 0$. Let $ {B_m}(\alpha )$ denote the subclass of $ B(\alpha )$ which is m-fold symmetric $ (f({e^{2\pi i/m}}z) = {e^{2\pi i/m}}f(z),m = 1,2, \cdots )$. Functions in $ B(\alpha )$ have been shown to be univalent. The authors obtain sharp coefficient inequalities for functions in $ {B_m}(1/N)$ where N is a positive integer. In addition an example of a Bazilevič function which is not close-to-convex is given.


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

  • [1] I. E. Bazilevič, On a case of integrability in quadratures of the Loewner-Kufarev equation, Mat. Sb. 37(79) (1955), 471-476. (Russian) MR 17, 356. MR 0072949 (17:356e)
  • [2] A. Bielecki and Z. Lewandowski, Sur un théorème concernant les fonctions univalentes linéairement asccessibles de M. Biernacki, Ann. Polon. Math. 12 (1962), 61-63. MR 26 #5151. MR 0147636 (26:5151)
  • [3] G. Birkhoff and G. Rota, Ordinary differential equations, 2nd, ed., Blaisdell, Waltham, Mass., 1969. MR 38 #4737.
  • [4] Z. Nehari, Conformal mapping, McGraw-Hill, New York, 1952. MR 13, 640. MR 0045823 (13:640h)
  • [5] C. Pommerenke, On the coefficients of close-to-convex functions, Michigan Math. J. 9 (1962), 259-269. MR 26 #5153. MR 0147638 (26:5153)
  • [6] -, Über die Subordination analytischer Funktionen, J. Reine Angew. Math. 218 (1965), 159-173. MR 31 #4900. MR 0180669 (31:4900)
  • [7] J. Zamorski, On Bazilevič schlicht functions, Ann. Polon. Math. 12 (1962), 83-90. MR 26 #1446. MR 0143896 (26:1446)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30.43

Retrieve articles in all journals with MSC: 30.43


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0283191-7
Keywords: Univalent, Bazilevič functions, close-to-convex functions, m-fold symmetric, Bieberbach conjecture, majorization
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society