Polynomially bounded solutions of the Loewner‎ ‎differential equation in several complex variables

Document Type: Research Paper

Authors

1 Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran.

2 Department of Mathematics, Faculty of Basic Science, University of Bonab, P.O. Box 5551-761167, Bonab, Iran.

3 Department of Mathematics‎, ‎Urmia University, Urmia‎, ‎Iran.

4 Department of Mathematics‎, ‎Rzesz'ow University of Technology‎, ‎Poland‎.

Abstract

‎We determine the‎ ‎form of polynomially bounded solutions to the Loewner differential ‎equation that is satisfied by univalent subordination chains of the‎ ‎form $f(z,t)=e^{\int_0^t A(\tau){\rm d}\tau}z+\cdots$‎, ‎where‎ ‎$A:[0,\infty]\rightarrow L(\mathbb{C}^n,\mathbb{C}^n)$ is a locally‎ ‎Lebesgue integrable mapping and satisfying the condition‎ ‎$$\sup_{s\geq0}\int_0^\infty\left\|\exp\left\{\int_s^t‎ ‎[A(\tau)-2m(A(\tau))I_n]\rm {d}\tau\right\}\right\|{\rm d}t<\infty,$$‎ ‎and $m(A(t))>0$ for $t\geq0$‎, ‎where‎ ‎$m(A)=\min\{\mathfrak{Re}\left\langle‎ ‎A(z),z\right\rangle:\|z\|=1\}$‎. ‎We also give sufficient conditions‎ ‎for $g(z,t)=M(f(z,t))$ to be polynomially bounded‎, ‎where $f(z,t)$ is‎ ‎an $A(t)$-normalized polynomially bounded Loewner chain solution to‎ ‎the Loewner differential equation and $M$ is an entire function‎. ‎On ‎the other hand‎, ‎we show that all $A(t)$-normalized polynomially‎ ‎bounded solutions to the Loewner differential equation are Loewner‎ ‎chains.‎

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Main Subjects


L. Arosio, F. Bracci, H. Hamada and G. Kohr, An abstract approach to Loewner chains, J. Anal. Math. 119 (2013) 89--114.

J. Becker, Lownersche differentialgleichung und schlichtheitskriterien, Math. Ann. 202 (1973) 321--335.

J. Becker, Uber die Losungsstruktur einer differentialgleichung in der konformen abbildung, J. Reine Angew. Math. 285 (1976) 66--74.

F. Bracci, M. D. Contreras and S. D. Madrigal, Evolution families and the Loewner equation II, Complex hyperbolic manifolds, Math. Ann. 344 (2009), no. 4, 947--962.

N. Dunford and J. T. Schwartz, Linear Operators, I, John Wiley & Sons, Inc., New York, 1988.

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.

M. Elin, S. Reich and D. Shoikhet, Complex dynamical systems and the geometry of domains in Banach spaces, Dissertationes Math. 427 (2004) 62 pages.

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 M. Kohr, Spirallike mappings and univalent subordination chains in Cn, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 (2008), no. 4, 717--740.

I. Graham, H. Hamada, G. Kohr and M. Kohr, Asymptotically spirallike mappings in several complex variables, J. Anal. Math. 105 (2008) 267--302.

I. Graham, H. Hamada, G. Kohr and M. Kohr, Parametric representation and asymptotic starlikeness in Cn, Proc. Amer. Math. Soc. 136 (2008), no. 11, 3963--3973.

I. Graham and G. Kohr, Geometric function theory in one and higher dimensions, Marcel Dekker, Inc., New York, 2003.

I. Graham, G. Kohr and M. Kohr, Loewner chains and prametric representation in several complex variables, J. Math. Anal. Appl. 281 (2003), no. 2, 425--438.

I. Graham, G. Kohr and J. A. Pfaltzgraff, The general solution of the Loewner differential equation on the unit ball in Cn, Contemp. Math. 382, Amer. Math. Soc., Providence, 2005.

S. Gong, Convex and starlike mappings in several complex variables, With a preface by David Minda. Mathematics and its Applications (China Series), 435, Kluwer Academic Publishers, Dordrecht, Science Press, Beijing, 1998.

K. E. Gustafson and O. K. M. Rao, Numerical range, The field of values of linear operators and matrices, Universitext, Springer-Verlag, New York, 1997.

H. Hamada, Polynomially bounded solutions to the Loewner differential equation in several complex variables, J. Math. Anal. Appl. 381 (2011), no. 1, 179--186.

H. Hmada and G. Kohr, Subordination chains and the growth theorem of spirallike mappings, Mathematica 42(65) (2000), no. 2, 153--161.

G. Kohr, Kernel convergence and biholomorphic mappings in several complex variables, Int. J. Math. Math. Sci. 2003 (2003), no. 67, 4229--4239.

E. Kubicka and T. Poreda, On the parametric representation of starlike maps of the unit ball in Cn into Cn, Demonstratio Math. 21 (1988), no. 2, 345--355.

J. A. Pfaltzgraff, Subordination chains and univalence of holomorphic mappings in Cn, Math. Ann. 210 (1974) 55--68.

J. A. Pfaltzgraff, Subordination chains and quasiconformal extension of holomorphic maps in Cn, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975) 13--25.

C. Pommereneke, Univalent functions, Vandenhoeck and Ruprecht, Gottingen, 1975

T. Poreda, On the univalent holomorphic maps of the unit polydisc in Cn which have the prametric representation, I- the geometrical properties, Ann. Univ. Mariae Curie Sk lodowska, Sect. A. 41 (1987) 105--113.

T. Poreda, On the univalent holomorphic maps of the unit polydisc in Cn which have the prametric representation, II- the necessary conditions and the sufficient conditions, Ann. Univ. Mariae Curie Sk lodowska, Sect. A. 41 (1987) 115--121.

T. Poreda, On the univalent subordination chains of holomorphic mappings in Banach spaces, Comment. Math. Prace Mat. 28 (1989), no. 2, 295--304.

T. Poreda, On generalized differential equations in Banach spaces, Dissertationes Math. 310 (1991) 50 pages.

S. Rahrovi, A. Ebadian and S. Shams, The non-normalized subordination chains with asymptotically spirallike mapping in several complex variables, General Math. 21 (2013), no. 2, 17--46.

S. Reich and D. Shoikhet, Nonlinear semigroups, Fixed points, and Geometry of Domains in Banach Spaces, Imperial College Press, London, 2005.

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 Math., 599, Springer, Berlin, 1977.

M. Voda, Solution of a Loewner chain equation in several complex variables, J. Math. Anal. Appl. 375 (2011), no. 1, 58--74.