Abstract: When lack of exact prior information, it is possible to use inequality constraints for improving solutions. To inequality constrained M-estimate (ICME), this paper derives the linear representations of the parameters estimator, residuals vector and adjusted vector of observations by using Bahadur linearization principle and aggregate function method. The corresponding variance-covariance matrix (VCV) and the mean squares error matrix (MSE) are also obtained. The Bahadur linear representation shows that the ICME is biased. The MSE matrix says the possibility that the solutions are improved, the condition of the improved solutions is that the maximum of the absolute inequality is smaller the derived threshold that is determined by the first and second nuisances, , ,of M-estimate and the nuisance ； To the least p-norm estimate in the case of Gauss distribution and p-norm distribution, the clear improvement conditions are given out. The derived forms and conclusions can be used for statistical analysis and inequality design.