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)

 
 

 

Distortion theorems for a special class of analytic functions


Author: Dorothy Browne Shaffer
Journal: Proc. Amer. Math. Soc. 39 (1973), 281-287
MSC: Primary 30A32
DOI: https://doi.org/10.1090/S0002-9939-1973-0315113-6
MathSciNet review: 0315113
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Sharp bounds are derived for the derivative of analytic functions of class $ {P_\alpha }$ defined by the condition $ \vert p(z) - 1/2\alpha \vert \leqq 1/2\alpha ,0 \leqq \alpha \leqq 1,p(0) = 1$. These results are improved for the class of functions with missing terms. Application is made to the class of functions with derivative $ \in {P_\alpha }$ and the radius of convexity is determined for this class.


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

  • [1] C. Carathéodory, Funktionentheorie. Band 2, Birkhäuser, Basel, 1950; English transl., Theory of functions of a complex variable. Vol. 2, Chelsea, New York, 1954. MR 12, 248; MR 16, 346.
  • [2] G. M. Goluzin, Geometric theory of functions of a complex variable, GITTL, Moscow, 1952; German transl., VEB Deutscher Verlag, Berlin, 1957; English transl., Transl. Math. Monographs, vol. 26, Amer. Math. Soc., Providence, R.I., 1969. MR 15, 112; MR 19, 125; MR 40 #308. MR 0247039 (40:308)
  • [3] W. Janowski, Extremal problems for a family of functions with positive real part and for some related families, Ann. Polon. Math. 23 (1970/71), 159-177. MR 42 #2005. MR 0267103 (42:2005)
  • [4] T. H. MacGregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc. 104 (1962), 532-537. MR 25 #4090. MR 0140674 (25:4090)
  • [5] -, A class of univalent functions, Proc. Amer. Math. Soc. 15 (1964), 311-317. MR 28 #2206. MR 0158985 (28:2206)
  • [6] Dorothy Browne Shaffer, On bounds for the derivative of analytic functions, Notices Amer. Math. Soc. 19 (1972), A-114. Abstract #691-30-13; Proc. Amer. Math. Soc. 37 (1973), 517-520. MR 0310256 (46:9357)

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/S0002-9939-1973-0315113-6
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society