Ebadian, A., Rahrovi, S., Shams, S., Sokol, J. (2016). Polynomially bounded solutions of the Loewner differential equation in several complex variables. Bulletin of the Iranian Mathematical Society, 42(3), 521-537.

A. Ebadian; S. Rahrovi; S. Shams; J. Sokol. "Polynomially bounded solutions of the Loewner differential equation in several complex variables". Bulletin of the Iranian Mathematical Society, 42, 3, 2016, 521-537.

Ebadian, A., Rahrovi, S., Shams, S., Sokol, J. (2016). 'Polynomially bounded solutions of the Loewner differential equation in several complex variables', Bulletin of the Iranian Mathematical Society, 42(3), pp. 521-537.

Ebadian, A., Rahrovi, S., Shams, S., Sokol, J. Polynomially bounded solutions of the Loewner differential equation in several complex variables. Bulletin of the Iranian Mathematical Society, 2016; 42(3): 521-537.

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

^{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.