On some convex combinations of biholomorphic mappings in several complex variables


In this paper, our interest is devoted to study the convex combinations of the form \((1-\lambda)f+\lambda g\) where \(\lambda \in(0,1)\) of biholomorphic mappings on the Euclidean unit ball \({\mathbb B}^n\) the case of several complex variables. Starting from a result proved by S. Trimble [26] and then extended by P.N. Chichra and R. Singh [3, Th. 2] which says that if f is starlike such that \(Re[f'(z)]>0\) then \((1-\lambda)z+\lambda f(z)\) is also starlike, we are interested to extend this result to higher dimensions.

In the first part of the paper, we construct starlike convex combinations using the identity mapping on \({\mathbb B}^n\) and some particular starlike mappings on \({\mathbb B}^n\).

In the second part of the paper, we define the class \(L_{\lambda}^∗B^n\) and prove results involving convex combinations of normalized locally biholomorphic mappings and Loewner chains. Finally, we propose a conjecture that generalize the result proved by Chichra and Singh.


Eduard Stefan Grigoriciuc
Babes-Bolyai” University, Faculty of Mathematics and Computer Science, Cluj-Napoca, Romania
”Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania


Biholomorphic mappings; Convex sums; Starlike mappings; Herglotz vector field; Loewner chains

Paper coordinates

E.S. Grigoriciuc, On some convex combinations of biholomorphic mappings in several complex variables, Filomat 36 (16) (2022), 5503-5519;


About this paper



Publisher Name

Faculty of Sciences and Mathematics, University of Nis, Serbia

Print ISSN


Online ISSN

google scholar link

[1] S.D. Bernardi, Convex and Starlike Univalent Functions, Trans. Amer. Math. Soc. 135 (1969), 429–446.
[2] D.M. Campbell, A survey of properties of the convex combination of univalent functions, Rocky Mountain J. Math. 5 (1975), 475–492.
[3] P. Chichra, R. Singh, Convex sum of univalent functions, J. Austral. Math. Soc. 14 (1972), 503–507.
[4] P.L. Duren, Univalent Functions, Springer-Verlag, New York, 1983.
[5] M. Elin, S. Reich, D. Shoikhet, Complex Dynamical Systems and the Geometry of Domains in Banach Spaces, Dissertationes Math. (Rozprawy Mat.) 427 (2004), 62 pp.
[6] I. Graham, H. Hamada, G. Kohr, Parametric Representation of Univalent Mappings in Several Complex Variables, Canad. J. Math. 54 (2002), 324–351.
[7] I. Graham, H. Hamada, G. Kohr, Radius problems for holomorphic mappings on the unit ball in Cn, Math. Nachr. 279 (2006), 1474–1490.
[8] I. Graham, H. Hamada, G. Kohr, M. Kohr, Extreme points, support points and the Loewner variation in several complex variables, Sci. China Math. 55(7) (2012), 1353–1366.
[9] I. Graham, G. Kohr, Geometric Function Theory in One and Higher Dimensions, Marcel Dekker Inc., New York, 2003.
[10] I. Graham, G. Kohr, M. Kohr, Loewner Chains and the Roper-Suffridge Extension Operator, J. Math. Anal. Appl. 247 (2000), 448–465.
[11] I. Graham, G. Kohr, M. Kohr, Loewner chains and parametric representation in several complex variables, J. Math. Anal. Appl. 281 (2003), 425–438.
[12] D.J. Hallenbeck, T.H. MacGregor, Linear Problems and Convexity Techniques In Geometric Function Theory, Pitman, Boston, 1984.
[13] H. Hamada, G. Kohr, Loewner chains and quasiconformal extension of holomorphic mappings, Ann. Polon. Math. 81 (2003), 85–100.
[14] H. Hamada, G. Kohr, Quasiconformal extension of biholomorphic mappings in several complex variables, J. Anal. Math. 96 (2005), 269–282.
[15] W.K. Hayman, Research Problems in Function Theory, The Athlone Press, London, 1967.
[16] G. Kohr, Basic Topics in Holomorphic Funcions of Several Complex Variables, Cluj University Press, Cluj-Napoca, 2003.
[17] T. Matsuno, On starlike theorems and convexlike theorems in the complex vector space, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 5 (1955), 88–95.
[18] T.H. MacGregor, The univalence of a linear combination of convex mappings, J. London Math. Soc. 44 (1969), 210–212.
[19] E.P. Merkes, On the convex sum of certain univalent functions and the identity function, Rev. Colombiana Math. 21 (1987), 5–12.
[20] J.A. Pfaltzgraff, Subordination Chains and Univalence of Holomorphic Mappings in Cn, Math. Ann. 210 (1974), 55–68.
[21] J.A. Pfaltzgraff, T.J. Suffridge, Close-to-starlike holomorphic functions of several complex variables, Pacif. J. Math. 57 (1975), 271–279.
[22] C. Pommerenke, Univalent Functions, Vandenhoeck and Ruprecht, Gottingen, 1975.
[23] S. Reich, D. Shoikhet, Nonlinear Semigroups, Fixed Points and the Geometry of Domains in Banach Spaces, World Scientific Publisher, Imperial College Press, London, 2005.
[24] K. Roper, T.J. Suffridge, Convexity properties of holomorphic mappings in Cn, Trans. Amer. Math. Soc., 351 (1999), 1803–1833.
[25] T.J. Suffridge, Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions, Lecture Notes in Math., 599 (1976), 146–159.
[26] S. Trimble, The convex sum of convex functions, Math. Z. 109 (1969), 112–114.


Related Posts