Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Regular functions $ f(z)$ for which $ z f'(z)$ is $ \alpha$-spiral


Authors: Richard J. Libera and Michael R. Ziegler
Journal: Trans. Amer. Math. Soc. 166 (1972), 361-370
MSC: Primary 30A32
DOI: https://doi.org/10.1090/S0002-9947-1972-0291433-2
MathSciNet review: 0291433
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A function $ f(z) = z + \Sigma _{n = 2}^\infty {a_n}{z^n}$ regular in the open unit disk $ \Delta = \{ z:\vert z\vert < 1\} $ is a (univalent) $ \alpha $-spiral function for real $ \alpha ,\vert\alpha \vert < \pi /2$, if $ \operatorname{Re} \{ {e^{i\alpha }}zf'(z)/f(z)\} > 0$ for z in $ \Delta $; in this case we write $ f(z) \in {\mathcal{F}_\alpha }$. A fundamental result of this paper shows that the transformation

$\displaystyle {f_ \ast }(z) = \frac{{azf((z + a)/(1 + \bar az))}}{{f(a)(z + a){{(1 + \bar az)}^{{e^{ - 2i\alpha }}}}}}$

defines a function in $ {\mathcal{F}_\alpha }$ whenever $ f(z)$ is in $ {\mathcal{F}_\alpha }$ and a is in $ \Delta $.

If $ g(z)$ is regular in $ \Delta ,g(0) = 0$ and $ g'(0) = 1$, then $ g(z)$ is in $ {\mathcal{G}_\alpha }$ if and only if $ zg'(z)$ is in $ {\mathcal{F}_\alpha }$. The main result of the paper is the derivation of the sharp radius of close-to-convexity for each class $ {\mathcal{G}_\alpha }$; it is given as the solution of an equation in r which is dependent only on $ \alpha $. (Approximate solutions of this equation were made by computer and these suggest that the radius of close-to-convexity of the class $ \mathcal{G} = { \cup _\alpha }{\mathcal{G}_\alpha }$ is approximately $ .99097^{+}$.) Additional results are also obtained such as the radius of convexity of $ {\mathcal{G}_\alpha }$, a range of $ \alpha $ for which $ g(z)$ in $ {\mathcal{G}_\alpha }$ is always univalent is given, etc. These conclusions all depend heavily on the transformation cited above and its analogue for $ {\mathcal{G}_\alpha }$.


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

  • [1] Hans Hornich, Ein Banachraum analytischer Funktionen in Zusammenhang mit den schlichten Funktionen, Monatsh. Math. 73 (1969), 36-45. MR 39 #4411. MR 0243087 (39:4411)
  • [2] -, Über einen Banachraum analytischer Funktionen, Manuscripta Math. 1 (1969), 79-86. MR 39 #4410. MR 0243086 (39:4410)
  • [3] W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J. 1 (1952), 169-185. MR 14, 966. MR 0054711 (14:966e)
  • [4] F. R. Keogh and E. P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20 (1969), 8-12. MR 38 #1249. MR 0232926 (38:1249)
  • [5] J. Krzyź, The radius of close-to-convexity within the family of univalent functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 10 (1962), 201-204. MR 0148887 (26:6384)
  • [6] R. J. Libera, Univalent $ \alpha $-spiral functions, Canad. J. Math. 19 (1967), 449-456. MR 35 #5599. MR 0214750 (35:5599)
  • [7] Z. Nehari, Conformal mapping, McGraw-Hill, New York, 1952. MR 13, 640. MR 0045823 (13:640h)
  • [8] -, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545-551. MR 10, 696. MR 0029999 (10:696e)
  • [9] M. S. Robertson, Radii of star-likeness and close-to-convexity, Proc. Amer. Math. Soc. 16 (1965), 847-852. MR 31 #5971. MR 0181744 (31:5971)
  • [10] -, Univalent functions $ f(z)$ for which $ zf'(z)$ is spirallike, Michigan Math. J. 16 (1969), 97-101. MR 39 #5785. MR 0244471 (39:5785)
  • [11] W. C. Royster, On the univalence of a certain integral, Michigan Math. J. 12 (1965), 385-387. MR 32 #1342. MR 0183866 (32:1342)
  • [12] R. Singh, A note on spiral-like functions, J. Indian Math. Soc. 33 (1969), 49-55. MR 41 #454. MR 0255794 (41:454)
  • [13] L. Špáček, Příspěek k teorii funkci prostyčh, Časopis Pěst. Mat. Fys. 62 (1933), 12-19.
  • [14] M. R. Ziegler, A class of regular functions related to univalent functions, Dissertation, University of Delaware, Newark, Del., 1970.

Similar Articles

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

Retrieve articles in all journals with MSC: 30A32


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0291433-2
Keywords: Spiral functions, univalence preserving transformations, radius of close-to-convexity
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society