Convex function (vector composition rule)
Solution 1:
A function $f$ is called log-convex if $\ln f$ is convex. It is not that difficult to show that a sum of two log-convex functions is log-convex. All you need to do is to notice that the function $\exp g_i$ is log-convex.
Another approach would be to show by definition for the case $m=2$ and then generalise to an arbitrary $m$.