Starlike functions
HTML articles powered by AMS MathViewer
- by Carl P. McCarty PDF
- Proc. Amer. Math. Soc. 43 (1974), 361-366 Request permission
Abstract:
Let ${\mathcal {S}^\ast }[\alpha ]$ denote the class of functions $f(z) = z + \sum \nolimits _{n = 2}^\infty {{a_n}{z^n}}$ analytic in $|z| < 1$ and for which $|zf’(z)/f(z) - 1| < 1 - \alpha$ for $|z| < 1$ and $\alpha \in [0,1)$. Sharp results concerning coefficients, distortion, and the radius of convexity are obtained. Furthermore, it is shown that $\sum \nolimits _{n = 2}^\infty {[(n - \alpha )/(1 - \alpha )]|{a_n}| < 1}$ is a sufficient condition for $f(z) \in {\mathcal {S}^\ast }[\alpha ]$.References
-
C. Carathéodory, Theory of functions of a complex variable. Vol. 2, Chelsea, New York, 1954. MR 16, 346.
- J. Clunie and F. R. Keogh, On starlike and convex schlicht functions, J. London Math. Soc. 35 (1960), 229–233. MR 110814, DOI 10.1112/jlms/s1-35.2.229
- A. W. Goodman, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc. 8 (1957), 598–601. MR 86879, DOI 10.1090/S0002-9939-1957-0086879-9
- Thomas H. MacGregor, A class of univalent functions, Proc. Amer. Math. Soc. 15 (1964), 311–317. MR 158985, DOI 10.1090/S0002-9939-1964-0158985-5
- Zeev Nehari, Conformal mapping, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1952. MR 0045823
- A. Schild, On a class of functions schlicht in the unit circle, Proc. Amer. Math. Soc. 5 (1954), 115–120. MR 60592, DOI 10.1090/S0002-9939-1954-0060592-3
- Albert Schild, On starlike functions of order $\alpha$, Amer. J. Math. 87 (1965), 65–70. MR 174729, DOI 10.2307/2373224
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 43 (1974), 361-366
- MSC: Primary 30A32
- DOI: https://doi.org/10.1090/S0002-9939-1974-0333147-3
- MathSciNet review: 0333147