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 certain analytic functions


Author: Hassoon S. Al-Amiri
Journal: Proc. Amer. Math. Soc. 42 (1974), 466-474
MSC: Primary 30A32
DOI: https://doi.org/10.1090/S0002-9939-1974-0330431-4
MathSciNet review: 0330431
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ F(z)$ be regular in the unit disk $ \Delta :\vert z\vert < 1$ and normalized by the conditions $ F(0) = 0$ and $ F'(0) = 1$. Let $ f(z) = \tfrac{1}{2}[zF(z)]'$. Recently Libera and Livingston have studied the mapping properties of $ f(z)$ when $ F(z)$ is known. In particular, they have determined the radius of starlikeness of order $ \beta $ for $ f(z)$ when $ F(z)$ is starlike of order $ \alpha ,0 \leqq \alpha \leqq \beta < 1$. The author extends this study to include the complementary case $ 0 \leqq \beta < \alpha $. Also, a different proof has been given to determine the disk in which $ \operatorname{Re} \{ f'(z)\} > \beta $ when $ \operatorname{Re} \{ F'(z)\} > \alpha ,0 \leqq \alpha < 1,0 \leqq \beta < 1$. All results are sharp.


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

  • [1] H. S. Al-Amiri, On the radius of univalence of certain classes of analytic functions, Colloq. Math. 28 (1973). MR 0328041 (48:6383)
  • [2] 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)
  • [3] R. J. Libera, Some radius of convexity problems, Duke Math. J. 31 (1964), 143-158. MR 28 #4099. MR 0160890 (28:4099)
  • [4] -, Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1965), 755-758. MR 31 #2389. MR 0178131 (31:2389)
  • [5] 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)
  • [6] 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)
  • [7] K. S. Padmanabhan, On the radius of univalence of certain classes of analytic functions, J. London Math. Soc. (2) 1 (1969), 225-231. MR 40 #331. MR 0247062 (40:331)
  • [8] M. S. Robertson, On the theory of univalent functions, Ann. of Math.(2) 37 (1936), 374-408. MR 1503286
  • [9] -, Variational methods for functions with positive real part, Trans. Amer. Math. Soc. 102 (1962), 82-93. MR 24 #A3288. MR 0133454 (24:A3288)
  • [10] V. A. Zmorovič, On bounds of convexity for starlike functions of order $ \alpha $ in the circle $ \vert z\vert < 1$ and in the circular region $ 0 < \vert z\vert < 1$, Mat. Sb. 68 (110) (1965), 518-526; English transl., Amer. Math. Soc. Transl. (2) 80 (1969), 203-213. MR 33 #5875. MR 0197712 (33:5875)

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/S0002-9939-1974-0330431-4
Keywords: Univalent functions, starlike functions of order $ \alpha $, radius of starlikeness of order $ \alpha $, functions with positive real part, extremal function
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society