Taiwanese Journal of Mathematics

JANOWSKI STARLIKENESS IN SEVERAL COMPLEX VARIABLES AND COMPLEX HILBERT SPACES

Paula Curt

Full-text: Open access

Abstract

In this paper, we consider two new subclasses, $S^*(a,b,B^n)$ and ${\mathcal A}S^*(a,b,B^n)$, of the class of starlike mappings on $B^n$ ($a,b\in \mathbb{R}$, $|a-1|\lt b\le a$, and $B^n$ is the Euclidean unit ball in $\mathbb{C}^n$). The class $S^*(a,b,B)$ is the $n$-dimensional version of Janowski class of one variable starlike functions. We obtain sharp growth results and upper distortion estimates for these two classes of starlike mappings. We also derive sufficient conditions for normalized holomorphic mappings (expressed in terms of their coefficient bounds) to belong to one of the classes $S^*(a,b,B^n)$, respectively ${\mathcal A}S^*(a,b,B^n)$. Finally, similar notions on the unit ball in a complex Hilbert space are analogously presented.

Article information

Source
Taiwanese J. Math., Volume 18, Number 4 (2014), 1171-1184.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706483

Digital Object Identifier
doi:10.11650/tjm.18.2014.3917

Mathematical Reviews number (MathSciNet)
MR3245436

Zentralblatt MATH identifier
1357.32012

Subjects
Primary: 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)

Keywords
biholomorphic mapping locally biholomorphic mapping starlike mapping almost starlike mapping

Citation

Curt, Paula. JANOWSKI STARLIKENESS IN SEVERAL COMPLEX VARIABLES AND COMPLEX HILBERT SPACES. Taiwanese J. Math. 18 (2014), no. 4, 1171--1184. doi:10.11650/tjm.18.2014.3917. https://projecteuclid.org/euclid.twjm/1499706483


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References

  • R. M. Ali, V. Ravichandran and N. Seenivasagan, Sufficient conditions for Janowski starlikeness, Int. J. Math. Math. Sci., 2007, Art. ID 62925, 7 pp.
  • P. Curt, A Marx-Strohhäcker theorem in several complex variables, Mathematica $($Cluj$)$, 39(62) (1997), 59-70.
  • S. Feng and K. Lu, The growth theorem for almost starlike mappings of order $\alpha $ on bounded starlike circular domains, Chinese Quart. J. Math., 15(2) (2000), 50-56.
  • S. X. Feng, Some Classes of Holomorphic Mappings in Several Complex Variables, University of Science and Technology of China, Doctor Thesis, 2004.
  • I. Graham, H. Hamada and G. Kohr, Parametric representation of univalent mappings in several complex variables, Canadian J. Math., 54 (2002), 324-351.
  • I. Graham and G. Kohr, Geometric Function Theory in One and Higher Dimensions, Marcel Dekker Inc., New York, 2003.
  • W. Janowski, Some extremal problems for certain families of analytic functions, I, Ann. Polon. Math., 28 (1973), 297-326.
  • G. Kohr, Certain partial differential inequalities and applications for holomorphic mappings defined on the unit ball of $\mathbb{C}^n$, Ann. Univ. Mariae Curie-Skl. Sect. A, 50 (1996), 87-94.
  • G. Kohr, On some sufficient conditions of almost starlikeness of order $\f{1}{2}$ in $\mathbb{C}^n$, Studia Univ. Babeş-Bolyai, Mathematica, 41, 3 (1996), 51-55.
  • G. Kohr, Using the method of Loewner chains to introduce some subclasses of univalent holomorphic mappings in $\mathbb{C}^n$, Rev. Roumaine Math. Pures Appl., 46 (2001), 743-760.
  • J. Liu, T. Liu and J. Wang, Distorsion theorems for subclasses of starlike mappings along a unit direction in $\mathbb{C}^n$, Acta Math. Sci., 32B(4) (2012), 1675-1680.
  • M. S. Liu and Y. C. Zhu, The radius of convexity and the sufficient condition for starlike mappings, B. Malays. Math. Sci. So., 35(2) (2012), 425-433.
  • T. Liu, J. Wang and J. Lu, Distortion theorems of starlike mappings in several complex variables, Taiwan. J. of Math., 15(6) (2011), 2601-2608.
  • J. A. Pfaltzgraff, Subordination chains and univalence of holomorphic mappings in $\mathbb{C}^n$, Math. Ann., 210 (1974), 55-68.
  • Y. Polatoglu and H. E. Ozkan, New subclasses of complex order, J. Prime Res. Math., 2 (2006), 157-169.
  • T. J. Suffridge, Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions, Lecture Notes Math., 599 (1976), 146-159.
  • H. Silverman, Subclasses of starlike functions, Rev. Roum. Math. Pures et Appl., 23 (1978), 1093-1099.
  • H. Silverman and E. M. Silvia, Subclasses of starlike functions subordinate to convex functions, Canad. J. Math., 37(1) (1985), 48-61.