Finding optimal dividend strategies is a classical problem in the financial and actuarial literature. The idea is that the company wants to pay some of its surplus as dividends, and the problem is to find a dividend strategy that maximizes the expected total discounted dividends received by the shareholders until ruin. Here we generalize results in [J. Paulsen, Adv. Appl. Probab., 39 (2007), pp. 669-689] in that the rate of growth of the surplus process is assumed to exceed the discounting factor whenever the surplus process is smaller than a fixed number (x)X In [J. Paulsen, Adv. Appl. Probab., 39 (2007), pp. 669-689] it was assumed that this rate of growth is always less than or equal to the discounting factor. It turns out that this generalization makes the problem much more complicated, and a simple barrier strategy is no longer always optimal.