Absence of Lavrentiev gap for non-autonomous functionals with (p, q)-growth

We consider non-autonomous functionals of the form F(u, Omega) =integral(Omega) f(x, Du(x))dx, where u : Omega -> R-N, Omega subset of R-n. We assume that f(x, z) grows at least as vertical bar z vertical bar(p) and at most as vertical bar z vertical bar(q). Moreover, f(x, z) is Holder continuous with respect to x and convex with respect to z. In this setting, we give a sufficient condition on the density f(x, z) that ensures the absence of a Lavrentiev gap.