We study the existence and shape preserving properties of a generalized Bernstein operator B n fixing a strictly positive function f 0, and a second function f 1 such that f 1/f 0 is strictly increasing, within the framework of extended Chebyshev spaces U n . The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator B n : C[a, b] → U n with strictly increasing nodes, fixing $${f_{0}, f_{1} \in U_{n}}$$ . If $${U_{n} \subset U_{n + 1}}$$ and U n+1 has a non-negative Bernstein basis, then there exists a Bernstein operator B n+1 : C[a, b] → U n+1 with strictly increasing nodes, fixing f 0 and f 1. In particular, if f 0, f 1, . . . , f n is a basis of U n such that the linear span of f 0, . . . , f k is an extended Chebyshev space over [a, b] for each k = 0, . . . , n, then there exists a Bernstein operator B n with increasing nodes fixing f 0 and f 1. The second main result says that under the above assumptions the following inequalities hold $$B_{n} f \geq B_{n+1} f \geq f$$ for all (f 0, f 1)-convex functions $${f \in C \, [ a, b]}$$ . Furthermore, B n f is (f 0, f 1)-convex for all (f 0, f 1)-convex functions $${f \in C \, [a, b]}$$ .