## Generating functions for some classes of univalent functions

HTML articles powered by AMS MathViewer

- by Zdzisław Lewandowski, Sanford Miller and Eligiusz Złotkiewicz PDF
- Proc. Amer. Math. Soc.
**56**(1976), 111-117 Request permission

## Abstract:

Let $P(z) = {e^{i\beta }} + {p_1}z + {p_2}{z^2} + \cdots$ be regular in the unit disc $\Delta$ with $|\beta | < \pi /2$, and let $\psi (u,v)$ be a continuous function defined in a domain of ${\mathbf {C}} \times {\mathbf {C}}$. With some very simple restrictions on $\psi (u,v)$ the authors prove a lemma that $\operatorname {Re} \psi (p(z),zp’(z)) > 0$ implies $\operatorname {Re} p(z) > 0$. This result is then used to generate subclasses of starlike, spirallike and close-to-convex functions.## References

- S. D. Bernardi,
*Convex and starlike univalent functions*, Trans. Amer. Math. Soc.**135**(1969), 429–446. MR**232920**, DOI 10.1090/S0002-9947-1969-0232920-2 - P. J. Eenigenburg, S. S. Miller, P. T. Mocanu, and M. O. Reade,
*On a subclass of Bazilevič functions*, Proc. Amer. Math. Soc.**45**(1974), 88–92. MR**344441**, DOI 10.1090/S0002-9939-1974-0344441-4 - I. S. Jack,
*Functions starlike and convex of order $\alpha$*, J. London Math. Soc. (2)**3**(1971), 469–474. MR**281897**, DOI 10.1112/jlms/s2-3.3.469 - Wilfred Kaplan,
*Close-to-convex schlicht functions*, Michigan Math. J.**1**(1952), 169–185 (1953). MR**54711** - Zdzisław Lewandowski, Sanford Miller, and Eligiusz Złotkiewicz,
*Gamma-starlike functions*, Ann. Univ. Mariae Curie-Skłodowska Sect. A**28**(1974), 53–58 (1976) (English, with Russian and Polish summaries). MR**412404** - R. J. Libera,
*Some classes of regular univalent functions*, Proc. Amer. Math. Soc.**16**(1965), 755–758. MR**178131**, DOI 10.1090/S0002-9939-1965-0178131-2 - Sanford S. Miller, Petru T. Mocanu, and Maxwell O. Reade,
*Bazilevič functions and generalized convexity*, Rev. Roumaine Math. Pures Appl.**19**(1974), 213–224. MR**338340**
L. Špǎcek,

*Contribution à la théorie des fonctions univalentes*, Čašopis. Pěst. Mat.

**62**(1933), 12-19.

## Additional Information

- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**56**(1976), 111-117 - DOI: https://doi.org/10.1090/S0002-9939-1976-0399438-7
- MathSciNet review: 0399438