In the estimation of a multivariate normal mean, it is shown that the problem of deriving shrinkage estimators improving on the maximum likelihood estimator can be reduced to that of solving an integral inequality. The integral inequality not only provides a more general condition than a differential inequality studied in the literature, but also handles non-differentiable or discontinuous estimators. The paper also gives characterization of prior distributions such that the resulting Bayes equivariant or generalized Bayes estimators are minimax. This characterization is provided by using the inverse Laplace transform. Finally, a simple proof for constructing a class of estimators improving on the James-Stein estimator is given based on the integral expression of the risk.