It is not convex on x>0. Consider a special case 1/log(1+1/(x^2)). When x->0, the second derivative < 0; When x->infinity, the second derivative > 0.
It is not convex on x>0. Consider a special case 1/log(1+1/(x^2)). When x->0, the second derivative < 0; When x->infinity, the second derivative > 0.