polynomially bounded solutions of the loewner‎ ‎differential equation in several complex variables

‎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_{sgeq0}int_0^inftyleft|expleft{int_s^t‎ ‎[a(tau)-2m(a(tau))i_n]rm {d}tauright}right|{rm d}t0$ for $tgeq0$‎, ‎where‎ ‎$m(a)=min{mathfrak{re}leftlangle‎ ‎a(z),zrightrangle:|z|=1}$‎. ‎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.‎