On Holomorphic Motions of $n$-Symmetric Functions

We generalize a problem examined by Duren on the univalence of a family of $n$-symmetric functions generated by integrals of functions of the form $\exp(\lambda \zeta^n)$. Our approach is based on the use of the inverse Faber transform, of the Martio–Sarvas univalence criterion, and of the $\lambda$-lemma of Mañé, Sad, and Sullivan. We also put forward a conjecture on the univalence of a family of $n$-symmetric functions, which is a weakened form of the Danikas–Ruscheweyh conjecture on the univalence of an integral transform of holomorphic functions.