Univalence of the generalized n-th root transform in Banach spaces

Determine whether, for a complex Banach space X with open unit ball B, the univalence of a normalized holomorphic mapping f in A(B) implies the univalence of its generalized n-th root transform g defined by g(x) = (K((x, e*)^n e))^{1/n} x on its domain B_p, where f(x) = K(x)x with K ∈ Hol(B, L(X)) and K(0) = Id, e ∈ OB is fixed, and e* ∈ J(e) is any support functional at e. Establish whether this implication holds in full generality beyond the one-dimensional-type case.

Background

In one complex variable, the classical n-th root transform preserves univalence: if f ∈ A(Δ) is univalent, then its root transform g is also univalent. The paper introduces a multidimensional analog for mappings on the open unit ball B of a complex Banach space by writing f(x) = K(x)x with K ∈ Hol(B, L(X)) and defining g(x) = (K((x, e*)^n e))^{1/n} x for fixed e ∈ OB and e* ∈ J(e).

Although g depends on the choice of K, the authors establish several transformation properties of this generalized root transform. They prove that the implication "f univalent ⇒ g univalent" holds when f is of one-dimensional type, but explicitly state uncertainty about whether it holds in the general multidimensional setting.