The GHS inequality and the Riemann hypothesis

Let V(t) be the even function on (-∞, ∞) which is related to the Riemann xi-function by Ξ(x/2)=4∫_-∞^∞ exp(ixt-V(t))dt. In a proof of certain moment inequalities which are necessary for the validity of the Riemann Hypothesis, it was previously shown that V'(t)/t is increasing on (0, ∞). We prove a stronger property which is related to the GHS inequality of statistical mechanics, namely that V' is convex on [0, ∞). The possible relevance of the convexity of V' to the Riemann Hypothesis is discussed.