Bonferroni-type inequalities; Chebyshev-type inequalities for the distributions on [0, n]

An elementary majorant-minorant method to construct the most stringent Bonferroni-type inequalities is presented. These are essentially Chebyshev-type inequalities for discrete probability distributions on the set {0, 1,..., n}, where n is the number of concerned events, and polynomials with specific properties on the set lead to the inequalities. All the known results are proved easily by this method. Further, the inequalities in terms of all the lower moments are completely solved by the method. As examples, the most stringent new inequalities of degrees three and four are obtained. Simpler expressions of Mrgritescu's inequality (1987, Stud. Cerc. Mat., 39, 246–251), improving Galambos' inequality, are given.