A property of Ditzian–Totik second order moduli

For a function f∈C2[0,1]f∈C2[0,1], we prove that limt→0+ωφ2(f,t)t2=‖φ2f″‖, where ωφ2(f,t) denotes a Ditzian–Totik-type modulus of order 2. We apply this result to obtain an asymptotic property for positive linear operators related to Voronovskaja type formulae.