Bivariate extension of the method of polynomials for Bonferroni-type inequalities (Q1343350)

The classical Bonferroni inequality is well-known. With the help of method of inclusion and exclusion a number of generalisations of this inequality were obtained [see, for example, \textit{R. M. Meyer}, Ann. Math. Stat. 40, 692-693 (1969; Zbl 0175.472)]. A special class of Bonferroni- type inequalities, so called bivariate inequalities, attracts attention. In recent joint publications (1992-1994) of the authors and M. Y. Lee there were obtained some bivariate inequalities by means of the method of polynomials, which in the univariate case is due to the first author and \textit{R. Mucci} [Publ. Math. 27, 263-268 (1980; Zbl 0455.60014)]. The authors of the present paper stress that their method of polynomials ``is not merely a method for proving inequalities but rather its real value is in the generation of new inequalities''. They illustrate their method by proving some new bivariate inequalities.