The Jacobian problem for one class of nonpolynomial mappings

The Jacobian conjecture in its classical form reads: If $ f: {{\Bbb R}}^n \rightarrow {{\Bbb R}}^n$ (or $ {{\Bbb C}}^n \rightarrow {{\Bbb C}}^n$) is a polynomial mapping with the Jacobian determinant $ J_f\ne 0$, then $ f $ is injective. This conjecture was first stated by Keller in 1939 for $n=2$ and disproved in the two-dimensional real case by Pinchuk in 1994. Since then the conjecture is formulated in modified form: If $ J_f\equiv \mathrm{const} \ne 0$ for a polynomial mapping $f$, then $f$ is injective. In 1998, this conjecture was included in the list of 18 mathematical problems of the forthcoming century. In this paper we describe a broad subclass of polynomial mappings where the classical conjecture is true; and we transfer these results to nonpolynomial mappings with $J_f\ne 0$.