Abstract In this paper, we establish some new Gronwall–Bellman-type delay dynamic inequalities on time scales, which can be used as a handy tool in the investigation of qualitative as well as quantitative analysis of solutions of delay dynamic equations on time scales. Our results generalize the main results of Ou-Iang [L. Ou-Iang, The boundedness of solutions of linear differential equations y″+A(t)y=0, Shuxue Jinzhan 3 (1957) 409-415], Jiang [F.C. Jiang, F.W. Meng, Explicit bounds on some new nonlinear integral inequality with delay, J. Comput. Appl. Math. 205 (2007) 479-486.] and Yuan [Z.L. Yuan, X.W. Yuan, F.W. Meng, Some new delay integral inequalities and their applications, Appl. Math. Comput. 208 (2009) 231-237].