OME 1986 Problem 4

Denote by $m(a,b)$ the arithmetic mean of positive real numbers $a,b$ . Given a positive real function $g$ having positive derivatives of the first and second order, define $\mu (a,b)$ the mean value of $a$ and $b$ with respect to $g$ by $2g(\mu (a,b)) = g(a)+g(b)$ . Decide which of the two mean values $m$ and $\mu$ is larger.