We determine the form of polynomially bounded solutions to
the Loewner differential equation that is satised 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