On the Optimality of Kounias-Sotirakglou Bounds

Bonferroni inequalities are significant in application for optimal allocation in tradable emission and other scary resources. Kounias and Sotirakglou improved Bonferroni inequalities, they give the lower and upper bounds for Bonferroni inequalities in the form of P（Ui=1^nAi）,but they did not discuss the optimal of lower and upper bounds for P（Ui=1^nAi）.In this paper, we give the method to obtain the optimum lower and upper bounds for Bonferroai inequalities in the form of ∑i=1^m（-1）^i＋1 aiSi for P（Ui=1^nAi）under the condition tha Q（r）≥（≤）0,The optimality of Kounias Sotirakglou bounds is also studied.

Bonferroni inequalities are significant in application for optimal allocation in tradable emission and other scary resources. Kounias and Sotirakglou improved Bonferroni inequalities,they give the lower and upper bounds for Bonferroni inequalities in the form of P(∪ni=1Ai),but they did not discuss the optimal of lower and upper bounds for P(∪ni=1Ai). In this paper, we give the method to obtain the optimum lower and upper bounds for Bonferroni inequalities in the form of ∑mi=1(-1)i+1aiSi for P(∪ni=1Ai) under the condition that Q(r)≥(≤)0. The optimality of Kounias-Sotirakglou bounds is also studied.