Pointwise simultaneous approximation by combinations of Bernstein operators

We prove in the present paper some new direct and inverse results on pointwise simultaneous approximation by combinations of Bernstein operators by using ωϕλ2r(f,t), where ωϕλ2r(f,t) is the Ditzian-Totik modulus of smoothness. We also give an equivalent relation on pointwise approximation by these operators using ωϕλ2r(f,t) when 0⩽λ<1-1/r.