Jacobi quasi-Nijenhuis algebroids *

In this paper, for a Jacobi algebroid (A, ρ), by introducing the notion of Jacobi quasi-Nijenhuis algebroids, which is a generalization of Poisson quasi-Nijenhuis manifolds introduced in [31] by Stiénon and Xu, we study generalized complex structures on the Courant-Jacobi algebroid A ⊕ A*, which unify generalized complex (contact) structures on an even(odd)-dimensional manifold.