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We study the existence and shape preserving properties of a generalizedBernstein operator $B_{n}$ fixing a strictly positive function $f_{0}$, and asecond function $f_{1}$ such that $f_{1}/f_{0}$ is strictly increasing, withinthe framework of extended Chebyshev spaces $U_{n}$. The first main result givesan inductive criterion for existence: suppose there exists a Bernstein operator$B_{n}:C[a,b]\to U_{n}$ with strictly increasing nodes, fixing $f_{0}, f_{1}\inU_{n}$. If $U_{n}\subset U_{n + 1}$ and $U_{n + 1}$ has a non-negativeBernstein basis, then there exists a Bernstein operator $B_{n+1}:C[a,b]\toU_{n+1}$ with strictly increasing nodes, fixing $f_{0}$ and $f_{1}.$ Inparticular, if $% f_{0},f_{1},...,f_{n}$ is a basis of $U_{n}$ such that thelinear 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 $B_{n}$ withincreasing nodes fixing $f_{0}$ and $f_{1}.$ The second main result says thatunder the above assumptions the following inequalities hold B_{n}f\geqB_{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 $f\in C[ a,b] .$ In the specific case of exponentialpolynomials we give alternative proofs of shape preserving properties bycomputing derivatives of the generalized Bernstein polynomials.

Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces