We determine the form of polynomially bounded solutions to the Loewner differential equation that is satis ed by univalent subordi- nation chains of the form f(z; t) = e ∫ t 0 A( )d z + , where A : [0;1] ! L(Cn;Cn) is a locally Lebesgue integrable mapping and satisfying the condition sup s0 ∫ 1 0 exp {∫ t s [A( ) 2m(A( ))In]d } dt < 1; and m(A(t)) > 0 for t 0, where m(A) = minfRe ⟨A(z); z⟩ : ∥z∥ = 1g. We also give sufficient conditions for g(z; t) = M(f(z; t)) to be polynomi- ally 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. Keywords: Biholomorphic
