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 radius of starlikeness of $ (zf)\sp{'} $ for $ f$ univalent


Author: Roger W. Barnard
Journal: Proc. Amer. Math. Soc. 53 (1975), 385-390
MSC: Primary 30A32
DOI: https://doi.org/10.1090/S0002-9939-1975-0382615-8
MathSciNet review: 0382615
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ S$ be the standard class of normalized univalent functions. For a given function $ f,\;f(z) = z + {a_2}{z^2} + \ldots $, regular for $ \vert z\vert < 1$, let $ r(f)$ be the radius of starlikeness of $ f$. In 1947, R. M. Robinson considered the combination $ {g_f}(z) = (zf)'/2$ for $ f\epsilon S$. He found a lower bound of .38 for $ r({g_f})$ for all $ f\epsilon S$. He noted that the standard Koebe function $ k,\;k(z) = z{(1 - z)^2}$, has its $ r({g_k})$ equal to $ 1/2$. A question that has been asked since Robinson's paper is whether $ 1/2$ is the minimum $ r({g_f})$ for all $ f$ in $ S$. It is shown here that this is not the case by giving examples of functions $ f$ whose $ r({g_{f}})$ is less than $ 1/2$.


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

  • [1] S. D. Bernardi, The radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 24 (1970), 312-318. MR 40 #4433. MR 0251202 (40:4433)
  • [2] S. D. Bernardi, The radius of univalence and starlikeness of certain analytic functions, Notices Amer. Math Soc. 20 (1973), A-332. Abstract #737-B135.
  • [3] G. Goluzin, Geometric theory of functions of a complex variable, GITTL, Moscow, 1952; English transl., Transl. Math. Monographs, vol. 26, Amer. Math. Soc., Providence, R. I., 1969. MR 15, 112; 40 #308. MR 0247039 (40:308)
  • [4] W. K. Hayman, Multivalent functions, Cambridge Tracts in Math. and Math. Phys., no. 48, Cambridge Univ. Press, Cambridge, 1958. MR 21 #7302. MR 0108586 (21:7302)
  • [5] J. A. Jenkins, On circularly symmetric functions, Proc. Amer. Math. Soc. 6 (1955), 620-624. MR 17, 249. MR 0072219 (17:249c)
  • [6] R. J. Libera and A. E. Livingston, On the univalence of some classes of regular functions, Proc. Amer. Math. Soc. 30 (1971), 327-336. MR 44 #5442. MR 0288244 (44:5442)
  • [7] A. E. Livingston, On the radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 17 (1966), 352-357. MR 32 #5861. MR 0188423 (32:5861)
  • [8] E. Netanyahu, On univalent functions in the unit disk whose image contains a given disk, J. Analyse Math. 23 (1970), 305-322. MR 43 #6420. MR 0280701 (43:6420)
  • [9] K. S. Padmanabhan, On the radius of univalence of certain classes of analytic functions, J. London Math. Soc. (2) 1 (1969), 225-331. MR 40 #331. MR 0247062 (40:331)
  • [10] R. M. Robinson, Univalent majorants, Trans. Amer. Math. Soc. 61 (1947), 1-35. MR 8, 370. MR 0019114 (8:370e)
  • [11] T. J. Suffridge, A coefficient problem for a class of univalent functions, Michigan Math. J. 16 (1969), 33-42. MR 39 #1646. MR 0240297 (39:1646)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30A32

Retrieve articles in all journals with MSC: 30A32


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0382615-8
Keywords: Univalent functions, starlike function, radius of starlikeness
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society