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

Graham, Ian; Hamada, Hidetaka; Kohr, Gabriela; Kohr, Mirela

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 4, pp. 717-740.

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.

Classification : 32H02, 30C45

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

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

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

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

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

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

[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 | Zbl

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