Taiwanese Journal of Mathematics

Extension Operators Preserving Janowski Classes of Univalent Functions

Andra Manu

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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., Volume 24, Number 1 (2020), 97-117.

Received: 15 May 2018
Revised: 12 March 2019
Accepted: 14 April 2019
First available in Project Euclid: 3 May 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH 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. 24 (2020), no. 1, 97--117. doi:10.11650/tjm/190407. https://projecteuclid.org/euclid.twjm/1556848821

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  • 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.