A property of Ditzian–Totik second order moduli

For a function f ∈ C 2 [ 0 , 1 ] , we prove that lim t → 0 + ω φ 2 ( f , t ) t 2 = ‖ φ 2 f ″ ‖ , where ω φ 2 ( 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.