## Taiwanese Journal of Mathematics

### ON SOME SUFFICIENT CONDITIONS FOR STARLIKENESS OF ORDER $\alpha$ IN $C^n$

#### Abstract

In this paper, we obtain some new sufficient conditions for starlikeness of order $\alpha$ of biholomorphic mappings on the unit ball in $C^n$ or a complex Hilbert space $X$ by using differential inequalities. We also obtain a distortion theorem and a covering theorem. As their special case, we obtain some sufficient conditions for starlikeness of order $\alpha$ of analytic functions on the unit disc in the complex plane $C$, which generalize some results of P. T. Mocanu and G. Oros.

#### Article information

Source
Taiwanese J. Math., Volume 10, Number 5 (2006), 1169-1182.

Dates
First available in Project Euclid: 20 July 2017

https://projecteuclid.org/euclid.twjm/1500557296

Digital Object Identifier
doi:10.11650/twjm/1500557296

Mathematical Reviews number (MathSciNet)
MR2253372

#### Citation

Liu, Ming-Sheng; Zhu, Yu-Can. ON SOME SUFFICIENT CONDITIONS FOR STARLIKENESS OF ORDER $\alpha$ IN $C^n$. Taiwanese J. Math. 10 (2006), no. 5, 1169--1182. doi:10.11650/twjm/1500557296. https://projecteuclid.org/euclid.twjm/1500557296

#### References

• P. Curt, A Marx-Strohhacker theorem on several complex variables, Mathematica $($Cluj$)$ 1 (1997), 59-70.
• G. Kohr, Certain partial differential inequalities and applications for holomorphic mappings defined on the unit ball of $C^n$, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 50 (1996), 87-94.
• P. Liczberski, Jack's lemma for holomorphic mappings in $C^n$, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 13 (1986), 131-140.
• M.-S. Liu, On certain sufficient condition for starlike functions, Soochow J. Math., 29 (2003), 407-412.
• S. S. Miller and P. T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. Appl., 65 (1978), 289-305.
• P. T. Mocanu, Some simple criteria for starlikeness and convexity, Libertas Math., 13 (1993), 27-40.
• P. T. Mocanu and G. Oros, A sufficient condition for starlikeness of order $\alpha$, Internat. J. Math. Math. Sci., 28 (2001), 557-560.
• T. J. Suffridge, Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions, Lecture Notes in Math., 599 (1975), 146-159.