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.

Receive Date: 29 October 2012,
Revise Date: 27 January 2015,
Accept Date: 07 February 2015

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.