Convexity of the gamma function with respect to Holder means

We prove that the gamma function is strictly multiplicatively concave on [0, x(0)] and strictly multiplicatively convex on [x(0), infinity], where x(0) is an element of [0, 1] is the unique solution of the equation psi(x) + xpsi'(x) = 0 and psi : [0, infinity] --> R is the digamma function: psi(x) = (dx)-(d) logGamma(x). Thus we answer to an open question raised by C. P. Niculescu [6, p. 164]. Likewise, we investigate the convexity of Gamma with respect to the Holder means.