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C. Loewner proved that the class of functions which have a parametric representation obtained by solving the Loewner differential equation with a driving term is dense in the class $$S$$ S of normalized univalent functions on the unit disc, because it contains all the single-slit mappings. I. Graham, H. Hamada, G. Kohr and M. Kohr suggested a generalization of this result to several complex variables, using control theory. We confirm this, namely we prove that the class of mappings which have an $$A$$ A -parametric representation obtained by solving the Loewner differential equation with infinitesimal generators which take values in the set of extreme points of the Carathéodory family in several complex variables is dense in the class of mappings with $$A$$ A -parametric representation, where $$A$$ A is a linear operator from $${\mathbb {C}}^n$$ C n to $${\mathbb {C}}^n$$ C n such that $$k_+(A)<2m(A)$$ k + ( A ) < 2 m ( A ) , $$k_+(A)$$ k + ( A ) is the Lyapunov index of $$A$$ A and $$m(A)=\min \{\mathfrak {R}\langle Az,z\rangle | z\in \mathbb {C}^n,\Vert z\Vert =1\}$$ m ( A ) = min { R ⟨ A z , z ⟩ | z ∈ C n , ‖ z ‖ = 1 } .
Computational Methods and Function Theory – Springer Journals
Published: Jan 8, 2015
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