关于Kantorovich型不等式

On Kantorovich?Type Inequality

给出了一种积分形式的Kantorovich型不等式为 :设a,A ,b,B和α均为正数 ,且a<A ,b <B .设E是可测集且μ(E) <+∞ .若p是一个在E上几乎处处为正的可积函数 ,f和g是在E上几乎处处为正的可测函数 ,且几乎处处有a≤f≤A ,b≤g≤B ,则∫Epfαdμ∫Epgαdμ ∫Ep(fg) α2 dμ2 ≤ 14ABabα4+ abABα42同时建立了等号成立的条件 .

This paper presents a Kantorovich?type inequality in an integral form as follows:Let a,A,b,B,α be positive numbers,a<A,b<B.Let E be a measurable set and μ(E)<+∞.If p is an integrable function and positive almost everyshere on E,f and g are measurable function and positive almost everywhere on E,and a≤f≤A,b≤g≤B,then∫ Epf α d μ∫ Epg α d μ∫ Ep(fg) α2 d μ 2≤14ABab α4 +abAB α4 2.Also,the condition of equality holding is established.