According to ito's lemma: $$\text{for}\ \ \ \ \ \ \ \ f=(t,X_t)\ , \ \ dX_t=\mu dt+\sigma dB_t $$ we get the formula: $$df = \left[\frac{\partial f}{\partial t}+\mu\frac{\partial f}{\partial x}+\frac12\sigma^2\frac{\partial^2 f}{\partial x^2}\right]dt\ +\ \frac{\partial f}{\partial x}dB_t $$
so the convexity has an impact on the drift term. If $\ f $ is a concave function of $X_t$, then $\frac{\partial^2 f}{\partial x^2}$ is negative, therefore the bigger the variation $\sigma^2$ is, the more it drags down the expectation of $\ f$ over time. Anyone has an intuitive explanation for that?