Spirallike mappings and univalent subordination chains in ${ℂ}^n$

In this paper we consider non-normalized univalent subordination chains and the connection with the Loewner differential equation on the unit ball in ${ℂ}^n$. To this end, we study the most general form of the initial value problem for the transition mapping, and prove the existence and uniqueness of solutions. Also we introduce the notion of generalized spirallikeness with respect to a measurable matrix-valued mapping, and investigate this notion from the point of view of non-normalized univalent subordination chains. We prove that such a spirallike mapping can be imbedded as the first element of a univalent subordination chain, and we present various particular cases and examples. If the matrix-valued mapping is constant, we obtain the usual notion of spirallikeness with respect to a linear operator.

[1] F. F. Bonsall and J. Duncan, “Numerical Ranges. II”, Cambridge Univ. Press, 1973. | MR 442682 | Zbl 0262.47001

[2] E. A. Coddington and N. Levinson, “Theory of Ordinary Differential Equations”, McGraw-Hill Book Co., New York-Toronto-London, 1955. | MR 69338 | Zbl 0064.33002

[3] Yu. L. Daleckii and M.G. Krein, “Stability of Solutions of Differential Equations in a Banach Space”, Translations of Mathematical Monographs, Vol. 43, American Mathematical Society, Providence, R.I., 1974. | MR 352639 | Zbl 0286.34094

[4] N. Dunford and J.T. Schwartz, “Linear Operators. I”, Interscience Publ., Inc., New York, 1966. | Zbl 0084.10402

[5] M. Elin, S. Reich and D. Shoikhet, Complex dynamical systems and the geometry of domains in Banach spaces, Dissertationes Math. 427 (2004), 1-62. | MR 2071666 | Zbl 1060.37038

[6] I. Graham, H. Hamada and G. Kohr, Parametric representation of univalent mappings in several complex variables, Canad. J. Math. 54 (2002), 324-351. | MR 1892999 | Zbl 1004.32007

[7] I. Graham, H. Hamada, G. Kohr and M. Kohr, Parametric representation and asymptotic starlikeness in ${ℂ}^n$, Proc. Amer. Math. Soc. 136 (2008), 267-302. | MR 2425737 | Zbl 1157.32016

[8] I. Graham, H. Hamada, G. Kohr and M. Kohr, Asymptotically spirallike mappings in several complex variables, J. Anal. Math. 105 (2008), 267-302. | MR 2438427 | Zbl 1148.32009

[9] I. Graham and G. Kohr, “Geometric Function Theory in One and Higher Dimensions”, Marcel Dekker Inc., New York, 2003. | MR 2017933 | Zbl 1042.30001

[10] I. Graham, G. Kohr and M. Kohr, Loewner chains and the Roper-Suffridge extension operator, J. Math. Anal. Appl. 247 (2000), 448-465. | MR 1769088 | Zbl 0965.32008

[11] I. Graham, G. Kohr and M. Kohr, Loewner chains and parametric representation in several complex variables, J. Math. Anal. Appl. 281 (2003), 425-438. | MR 1982664 | Zbl 1029.32004

[12] K. Gurganus, $\Phi$-like holomorphic functions in ${ℂ}^n$ and Banach spaces, Trans. Amer. Math. Soc. 205 (1975), 389-406. | MR 374470 | Zbl 0299.32018

[13] K. E. Gustafson and D.K.M. Rao, “Numerical Range. The Field of Values of Linear Operators and Matrices”, Springer-Verlag, New York, 1997. | MR 1417493 | Zbl 0874.47003

[14] H. Hamada and G. Kohr, Subordination chains and the growth theorem of spirallike mappings, Mathematica (Cluj) 42 (65) (2000), 153-161. | MR 1988620 | Zbl 1027.46094

[15] H. Hamada and G. Kohr, An estimate of the growth of spirallike mappings relatve to a diagonal matrix, Ann. Univ. Mariae Curie-Skłodowska, Sect. A. 55 (2001), 53-59. | MR 1845250 | Zbl 1018.32007

[16] L. Harris, The numerical range of holomorphic functions in Banach spaces, Amer. J. Math. 93 (1971), 1005-1019. | MR 301505 | Zbl 0237.58010

[17] L. A. Harris, S. Reich and D. Shoikhet, Dissipative holomorphic functions, Bloch radii, and the Schwarz lemma, J. Anal. Math. 82 (2000), 221-232. | MR 1799664 | Zbl 0972.46029

[18] G. Kohr, Using the method of Löwner chains to introduce some subclasses of biholomorphic mappings in ${ℂ}^n$, Rev. Roumaine Math. Pures Appl. 46 (2001), 743-760. | MR 1929522 | Zbl 1036.32014

[19] J. A. Pfaltzgraff, Subordination chains and univalence of holomorphic mappings in ${ℂ}^n$, Math. Ann. 210 (1974), 55-68. | MR 352510 | Zbl 0275.32012

[20] J. A. Pfaltzgraff and T.J. Suffridge, An extension theorem and linear invariant families generated by starlike maps, Ann. Univ. Mariae Curie-Skłodowska, Sect. A. 53 (1999), 193-207. | MR 1778828 | Zbl 0996.32006

[21] C. Pommerenke, Über die subordination analytischer funktinonen, J. Reine Angew. Math. 218 (1965), 159-173. | MR 180669 | Zbl 0184.30601

[22] C. Pommerenke, “Univalent functions”, Vandenhoeck & Ruprecht, Göttingen, 1975. | Zbl 0298.30014

[23] T. Poreda, On the univalent holomorphic maps of the unit polydisc in ${ℂ}^n$ which have the parametric representation, I-the geometrical properties, Ann. Univ. Mariae Curie-Skłodowska, Sect. A. 41 (1987), 105-113. | MR 1049182 | Zbl 0698.32004

[24] T. Poreda, On the univalent holomorphic maps of the unit polydisc in ${ℂ}^n$ which have the parametric representation, II-the necessary conditions and the sufficient conditions, Ann. Univ. Mariae Curie-Skłodowska, Sect. A. 41 (1987), 115-121. | MR 1049183 | Zbl 0698.32005

[25] T. Poreda, On the univalent subordination chains of holomorphic mappings in Banach spaces, Comment. Math. Prace Mat. 28 (1989), 295-304. | MR 1024945 | Zbl 0694.46033

[26] T. Poreda, On generalized differential equations in Banach spaces, Dissertationes Math. 310 (1991), 1-50. | MR 1104523 | Zbl 0745.35048

[27] S. Reich and D. Shoikhet, “Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces”, Imperial College Press, London, 2005. | MR 2022955 | Zbl 1089.46002

[28] K. Roper and T. J. Suffridge, Convex mappings on the unit ball of ${ℂ}^n$, J. Anal. Math. 65 (1995), 333-347. | MR 1335379 | Zbl 0846.32006

[29] T. J. Suffridge, Starlike and convex maps in Banach spaces, Pacific J. Math. 46 (1973), 575-589. | MR 374914 | Zbl 0263.30016

[30] T. J. Suffridge, Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions, In: “Lecture Notes in Math.”, Springer-Verlag 599 (1977), 146-159. | MR 450601 | Zbl 0356.32004

[31] K. Yosida, “Functional Analysis”, Springer-Verlag, 1965.