Taiwanese Journal of Mathematics

Extension Operators Preserving Janowski Classes of Univalent Functions

Andra Manu

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

Full-text: Open access


In this paper, our main interest is devoted to study the extension operator $\Phi_{n,\alpha,\beta} \colon \mathcal{L}S \to \mathcal{L}S_n$ given by $\Phi_{n,\alpha,\beta}(f)(z) = \big( f(z_1), \widetilde{z}(f(z_1)/z_1)^{\alpha} (f'(z_1))^{\beta} \big)$, $z = (z_1,\widetilde{z}) \in \mathbf{B}^n$, where $\alpha,\beta \geq 0$. We shall prove that if $f \in S$ can be embedded as the first element of a $g$-Loewner chain with $g \colon U \to \mathbb{C}$ given by $g(\zeta) = (1+A\zeta)/(1+B\zeta)$, $|\zeta| \lt 1$, and $-1 \leq B \lt A \leq 1$, then $F = \Phi_{n,\alpha,\beta}(f)$ can be embedded as the first element of a $g$-Loewner chain on the unit ball $\mathbf{B}^n$ for $\alpha \in [0,1]$, $\beta \in [0,1/2]$ and $\alpha + \beta \leq 1$. As a consequence, the operator $\Phi_{n,\alpha,\beta}$ preserves the notions of Janowski starlikeness on $\mathbf{B}^n$ and Janowski almost starlikeness on $\mathbf{B}^n$. Particular cases will be also mentioned.

On the other hand, we are also concerned about some radius problems related to the operator $\Phi_{n,\alpha,\beta}$ and the Janowski class $S^*(a,b)$. We compute the radius $S^*(a,b)$ of the class $S$ (respectively $S^*$).

Article information

Taiwanese J. Math., Advance publication (2019), 21 pages.

First available in Project Euclid: 3 May 2019

Permanent link to this document

Digital Object Identifier

Primary: 32H99: None of the above, but in this section
Secondary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)

$g$-Loewner chain $g$-parametric representation $g$-starlikeness $g$-spirallikeness Janowski starlikeness Janowski almost starlikeness Roper-Suffridge extension operator


Manu, Andra. Extension Operators Preserving Janowski Classes of Univalent Functions. Taiwanese J. Math., advance publication, 3 May 2019. doi:10.11650/tjm/190407. https://projecteuclid.org/euclid.twjm/1556848821

Export citation


  • T. Chirilă, An extension operator associated with certain $g$-Loewner chains, Taiwanese J. Math. 17 (2013), no. 5, 1819–1837.
  • ––––, Subclasses of biholomorphic mappings associated with $g$-Loewner chains on the unit ball in $\mathbb{C}^n$, Complex Var. Elliptic Equ. 59 (2014), no. 10, 1456–1474.
  • P. Curt, A Marx-Strohhäcker theorem in several complex variables, Mathematica 39(62) (1997), no. 1, 59–70.
  • ––––, Janowski starlikeness in several complex variables and complex Hilbert spaces, Taiwanese J. Math. 18 (2014), no. 4, 1171–1184.
  • P. Duren, I. Graham, H. Hamada and G. Kohr, Solutions for the generalized Loewner differential equation in several complex variables, Math. Ann. 347 (2010), no. 2, 411–435.
  • 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, Canad. J. Math. 54 (2002), no. 2, 324–351.
  • I. Graham, H. Hamada, G. Kohr and T. J. Suffridge, Extension operators for locally univalent mappings, Michigan Math. J. 50 (2002), no. 1, 37–55.
  • I. Graham and G. Kohr, An extension theorem and subclasses of univalent mappings in several complex variables, Complex Var. Theory Appl. 47 (2002), no. 1, 59–72.
  • ––––, Geometric Function Theory in One and Higher Dimensions, Monographs and Textbooks in Pure and Applied Mathematics 255, Marcel Dekker, New York, 2003.
  • I. Graham, G. Kohr and M. Kohr, Loewner chains and the Roper-Suffridge extension operator, J. Math. Anal. Appl. 247 (2000), no. 2, 448–465.
  • ––––, Loewner chains and parametric representation in several complex variables, J. Math. Anal. Appl. 281 (2003), no. 2, 425–438.
  • H. Hamada and T. Honda, Sharp growth theorems and coefficient bounds for starlike mappings in several complex variables, Chin. Ann. Math. Ser. B 29 (2008), no. 4, 353–368.
  • H. Hamada and G. Kohr, Subordination chains and the growth theorem of spirallike mappings, Mathematica 42(65) (2000), no. 2, 153–161.
  • H. Hamada, G. Kohr and M. Kohr, Parametric representation and extension operators for biholomorphic mappings on some Reinhardt domains, Complex Var. Theory Appl. 50 (2005), no. 7-11, 507–519.
  • 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-Skłodowska Sect. A 50 (1996), 87–94.
  • ––––, On some sufficient conditions of almost starlikeness of order $1/2$ in $C^n$, Studia Univ. Babeş-Bolyai Math. 41 (1996), no. 3, 51–55.
  • X. Liu, The generalized Roper-Suffridge extension operator for some biholomorphic mappings, J. Math. Anal. Appl. 324 (2006), no. 1, 604–614.
  • X. S. Liu and T. S. Liu, The generalized Roper-Suffridge extension operator for spirallike mappings of type $\beta$ and order $\alpha$, Chinese Ann. Math. Ser. A 27 (2006), no. 6, 789–798.
  • J. R. Muir, Jr., Extension of convex mappings of order $\alpha$ of the unit disk in $\mathbb{C}$ to convex mappings of the unit ball in $\mathbb{C}^n$, J. Math. Anal. Appl. 356 (2009), no. 1, 369–377.
  • M. M. Nargesi, R. M. Ali and V. Ravichandran, Radius constants for analytic functions with fixed second coefficient, The Scientific World Journal 2014 (2014), Art. ID 898614, 6 pp.
  • J. A. Pfaltzgraff, Subordination chains and univalence of holomorphic mappings in $C^n$, Math. Ann. 210 (1974), 55–68.
  • C. Pommerenke, Univalent Functions, Studia Mathematica/Mathematische Lehrbücher, Band XXV, Vandenhoeck & Ruprecht, Göttingen, 1975.
  • K. A. Roper and T. J. Suffridge, Convex mappings on the unit ball of $C^n$, J. Anal. Math. 65 (1995), 333–347.
  • H. Silverman, Subclasses of starlike functions, Rev. Roumaine Math. Pures Appl. 23 (1978), no. 7, 1093–1099.
  • H. Silverman and E. M. Silvia, Subclasses of starlike functions subordinate to convex functions, Canad. J. Math. 37 (1985), no. 1, 48–61.
  • T. J. Suffridge, Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions, Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), 146–159, Lecture Notes in Mathematics 599, Springer, Berlin, 1977.
  • M. I. Voda, Loewner Theory in Several Complex Variables and Related Problems, Thesis (Ph.D.)–University of Toronto (Canada), 2011.
  • C. Wang, Y. Cui and H. Liu, New subclasses of biholomorphic mappings and the modified Roper-Suffridge operator, Chin. Ann. Math. Ser. B 37 (2016), no. 5, 691–704.
  • Q.-H. Xu and T.-S. Liu, Löwner chains and a subclass of biholomorphic mappings, J. Math. Anal. Appl. 334 (2007), no. 2, 1096–1105.
  • Y.-C. Zhu and M.-S. Liu, The generalized Roper-Suffridge extension operator on Reinhardt domain $D_p$, Taiwanese J. Math. 14 (2010), no. 2, 359–372.