A Leibniz differentiation formula for positive operators

Is is shown that for n→+∞the Leibnizian combination L n(fg)−fL n(g)−gL n(f ) converges uniformly to zero on a compact interval W if the positive operators Ln belong to a certain class (including Bernstein, Gauss–Weierstrass and many others), and if the moduli of continuity of f, g satisfy ωW(f ; h)ωW(g; h) = o(h) as h→ 0+. A counterexample shows that Lipschitz conditions are not appropriate to bring about a second-order version of this formula.  2002 Elsevier Science (USA). All rights reserved.