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


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