Convexity preserving and predicting by Bernstein polynomials

Abstract It is known that the Bernstein polynomials of a function f defined on [0, 1 ] preserve its convexity properties, i.e., if f ( n ) ⩾ 0 then for m ⩾ n , ( B m f ) ( n ) ⩾ 0. Moreover, if f is n -convex then ( B m f ) ( n ) ⩾ 0. While the converse is not true, we show that if f is bounded on ( a , b ) and if for every subinterval [ α , β ] ⊂ ( a , b ) the n th derivative of the m th Bernstein polynomial of f on [α, β] is nonnegative then f is n -convex.