Embedding theorems for holomorphic functions of several complex variables

• Autorzy: Długosz R
• Języki publikacji: EN

Abstrakty

EN

Many authors studied families ^XG of complex valued functions, which are holomorphic in bounded complete n-circular domains G⊂Cⁿ and fulfill some geometric conditions. The above functions were applied later to research families of locally biholomorphic mappings in Cⁿ. In this paper we consider a problem of inclusions between a few of such families ^XG and families M^K_G, k=2, 3, …, which are defined by applying a function decomposition with respect to the group of kth roots of unity.

Słowa kluczowe

EN

holomorphic functions of several complex variables; n-circular domains in Cn; Minkowski function; function decomposition with respect to a cyclic group; embedding theorem

PL

funkcje holomorficzne; zmienne zespolone; funkcja Minkowskiego; twierdzenie o zanurzaniu

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2013
• Tom: Vol. 19, nr 1
• Strony: 153--165
• Opis fizyczny: Bibliogr. 25 poz.

Twórcy

autor

Długosz R

• Institute of Mathematics, Lodz University of Technology, ul. Wólczańska 215, 90-924 Łódź, Poland

Bibliografia

• [1] J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann. Math. 17 (1915/16), 12-22.
• [2] I. I. Bavrin, A Class of Regular Bounded Functions in the Case of Several Complex Variables and Extreme Problems in That Class (in Russian), Moskov Obi. Ped. Inst., Moscov, 1976.
• [3] R. Długosz and E. Leś, Embedding theorems and extreme problems for holomorphic functions on circular domains of Cn, Complex Van Elliptic Equ., to appear.
• [4] K. Dobrowolska and P. Liczberski, On some differential inequalities for holomorphic functions of many variables, Demonstratio Math. 14 (1981), 383-398.
• [5] I. Dziubiński and R. Sitarski, On classes of holomorphic functions of many variables starlike and convex on some hypersurfaces, Demonstratio Math. 13 (1980), 619-632.
• [6] S. Fukui, On the estimates of coefficients of analytic functions, Sci. Rep. Tokyo Kyoiku Daigaku 10 (1969), 216-218.
• [7] H. Hamada, T. Honda and G. Kohr, Parabolic starlike mappings in several complex variables, Manuscripta Math. 123 (2007), 301-324.
• [8] T. Higuchi, On coefficients of holomorphic functions of several complex variables, Sci. Rep. Tokyo Kyoiku Daigaku 8 (1965), 251-258.
• [9] Z. Jakubowski and J. Kamiński, On some classes of Mocanu-Bazylevich functions, Acta Univ. Lodz, Folia Math. 5 (1992), 39-62.
• [10] W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J. 1 (1952), 169-185.
• [11] Z. Lewandowski, Sur l’identité decertaines classes de fonctions univalentes I, II, Ann. Univ. Mariae Curie-Skłodowska, Sect A 12 (1958), 131-146,14 (1960), 19-16.
• [12] P. Liczberski, On the subordination of holomorphic mappings in Cn, Demonstratio Math. 2 (1986), 293-301.
• [13] P. Liczberski and J. Połubiński, On (j,k)-symmetrical functions, Math. Bohem. 120 (1995), 13-28.
• [14] P. Liczberski and J. Połubiński, Symmetrical series expansion of complex valued functions, New Zealand J. Math. 27 (1998), 245-253.
• [15] P. Liczberski and J. Połubiński, A uniqueness theorem of Cartan-Gutzmer type for holomorphic mappings in Cn, Ann. Polon. Math. 79 (2002), 121-127.
• [16] P. Liczberski and J. Połubiński, Functions (j,k)-symmetrical and functional equations with iterates of the unknown function, Publ. Math. Debrecen 60 (2002), 291-305.
• [17] A. Marchlewska, On certain subclasses of Bawrin’s families of holomorphic maps of two complex variables, in: Proceedings of the Fifth Environmental Mathematical Conference (Rzeszów-Lublin-Lesko 1998), Press of Catholic University of Lublin, Lublin (1999), 99-106.
• [18] A. Marchlewska, Ona generalization of close-to-convexity for complex holomorphic functions in Cn, Demonstratio Math. 4 (2005), 847-856.
• [19] Y. Michiwaki, Note on some coefficients in a starlike functions of two complex variables, Res. Rep. Nagaoka Tech. College 1 (1963), 151-153.
• [20] I. R. Nezhmetdinov and S. Ponnusamy, On the class of univalent functions starlike with respect to N-symmetric points, Hokkaido Math. J. 31 (2002), 61-77.
• [21] J. A. Pfaltzgraff and T. J. Suffridge, An extension theorem and linear invariant families generated by starlike maps, Ann. UMCS Sect. Math. 53 (1999), 193-207.
• [22] C. Pommerenke, Univalent Functions, Vandenhoeck and Ruprecht, Göttingen, 1975.
• [23] K. Sakaguchi, On certain univalent mapping, J. Math. Soc. Japan 11 (1959), 72-75.
• [24] J. Stankiewicz, Functions of two complex variables regular in half-space, Folia Sci. Univ. Techn. Rzeszów Math. 19 (1996), 107-116.
• [25] E. Study, Vorlesungen über ausgewählte Gegenstände der Geometrie, Teubner, Heft 2, Leipzig, 1913.