Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

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) is an element of U-n subset of If U-n subset of Un+ 1 and Un+ 1 has a non-negative Bernstein basis, then there exists a Bernstein operator Bn+ 1 : C[a, b] -> Un+ 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