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
