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
∫ t
0 A(τ)dτ
z + · · · , where A : [0,∞] →
L(Cn, Cn) is a locally Lebesgue integrable mapping and satisfying the
condition
sup
s≥0
∫ ∞
0
exp {∫ t
s
[A(τ) − 2m(A(τ))In]dτ
}
dt < ∞,
and m(A(t)) > 0 for t ≥ 0, where m(A) = min{Re ⟨A(z), z⟩ : ∥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