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)

 
 

 

The univalence of an integral


Author: W. M. Causey
Journal: Proc. Amer. Math. Soc. 27 (1971), 500-502
MSC: Primary 30.42
DOI: https://doi.org/10.1090/S0002-9939-1971-0280700-9
MathSciNet review: 0280700
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ f(z)$ be a normalized function, analytic and univalent in the open unit disc. It is shown that if $ g(z) = \int_0^z {{{(f(t)/t)}^\alpha }dt} $, then $ g$ is univalent in the open unit disc if $ \alpha $ is a complex number satisfying $ 0 \leqq \vert\alpha \vert \leqq (\surd 2 - 1)/4$.


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

  • [1] W. M. Causey, The close-to-convexity and univalence of an integral, Math. Z. 99 (1967), 207-212. MR 35 #6807. MR 0215972 (35:6807)
  • [2] P. L. Duren, H. S. Shapiro and A. L. Shields, Singular measures and domains not of the Smirnov type, Duke Math. J. 33 (1966), 247-254. MR 33 #7506. MR 0199359 (33:7506)
  • [3] Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545-551. MR 10, 696. MR 0029999 (10:696e)
  • [4] M. Nunokawa, On the univalency and multivalency of certain analytic functions, Math. Z. 104 (1968), 394-404. MR 37 #2971. MR 0227387 (37:2971)
  • [5] -, On the univalence of a certain integral, Trans. Amer. Math. Soc. 146 (1969), 439-446. MR 40 #4436. MR 0251205 (40:4436)
  • [6] W. C. Royster, On the univalence of a certain integral, Michigan Math. J. 12 (1965), 385-387. MR 32 #1342. MR 0183866 (32:1342)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30.42

Retrieve articles in all journals with MSC: 30.42


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0280700-9
Keywords: Schwarzian derivative, Poisson integral, univalent functions
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society