In this paper, we address an online knapsack problem under a convex size profit function. We first give a greedy online algorithm with a competitive ratio 2. Then we propose an improved online algorithm with a competitive ratio 5/3. We also prove that when the convex function has a specific property, our improved online algorithm is (1 + root 5)/2-competitive, which is optimal. Finally, we prove that the lower bound of this problem is (1 + root 5)/2. (C) 2013 Elsevier B.V. All rights reserved.