A new representation and shape-preserving properties of perturbed Bernstein operators

We introduce a new representation of recently introduced perturbed Bernstein operators BnM$$ {B}_n circumflex M $$ in terms of classical Bernstein operators. Using this new representation, we investigate the shape-preserving properties of the operator BnM$$ {B}_n circumflex M $$. In particular, we prove that perturbed Bernstein operators preserve monotonicity and convexity of functions for certain cases. On the other hand, we demonstrate with some counterexamples that monotonicity and convexity preserving properties fail in other cases. Moreover, we present some weaker results in these cases.