A refinement of the variation diminishing property of Bézier curves

For a given polynomial , expressed in the Bernstein basis over an interval [a,b], we prove that the number of real roots of F(t) in [a,b], counting multiplicities, does not exceed the sum of the number of real roots in [a,b] of the polynomial (counting multiplicities) with the number of sign changes in the two sequences (p0,…,pk) and (pl,…,pn) for any value k,l with 0⩽k⩽l⩽n. As a by product of this result, we give new refinements of the classical variation diminishing property of Bézier curves.