Sequences of non-decreasing (non-increasing) lower (upper) bounds for the renewal-type equation as well as for the renewal function which are improvements of the famous corresponding bounds of Marshal [(1973). Linear bounds on the renewal function. SIAM Journal on Applied Mathematics 24(2): 245–250] are given. Also, sequences such bounds converging to the ordinary renewal function are obtained for several reliability classes of the lifetime distributions of the inter-arrival times, which are refinements of all of the existing known corresponding bounds. For the first time, a lower bound for the renewal function with DMRL lifetimes is given. Finally, sequences of such improved bounds are given for the ordinary renewal density as well as for the right-tail of the distribution of the forward recurrence time.

更新型方程和更新函数的非递减（非递增）下（上）界序列是对 Marshal [(1973) 著名的对应界的改进。更新函数的线性界限。SIAM 应用数学杂志24(2): 245–250] 给出。此外，对于到达间隔时间的寿命分布的几个可靠性等级，获得了收敛于普通更新函数的这样的边界的序列，其是所有现有的已知对应边界的细化。首次给出了具有 DMRL 生命周期的更新函数的下限。最后，针对普通更新密度以及前向递归时间分布的右尾给出了这种改进边界的序列。