积分不等式(Ⅰ)

本文最初欲把Ｂｅｌｍａｎ不等式推广成：已知φ（ｘ）≤Ｃ（ｘ）＋Ｋ１（ｘ）∫ｘａＨ１（ζ）φ（ζ）ｄζ＋Ｋ２（ｘ）∫ｘａＨ２（ζ）φ（ζ）ｄζ（其中：Ｋｉ（ｘ）≥０,Ｈｉ（ｘ）≥０,ｉ＝１,２）,求适合上述不等式的φ（ｘ）的最优上界Ψ（ｘ）（ｘ≥ａ）。但后来证明这个最优上界Ψ（ｘ）是不能用初等方法求出的,只知道Ψ（ｘ）是存在的且适合积分方程：Ψ（ｘ）＝Ｃ（ｘ）＋Ｋ１（ｘ）∫ｘａＨ１（ζ）Ψ（ζ）ｄζ＋Ｋ２（ｘ）∫ｘａＨ２（ζ）Ψ（ζ）ｄζ。把此结论加以全面的推广即得到本文在高维向量空间中的多变量线性积分不等式

Integral Inequality (I)

The main conclusion of the paper is as fallows: Theorem: suppose x=(x 1,x 2,…,x m),ξ=(ξ 1,ξ 2,…,ξ m) are points of m-dimension euclidean space E m,A(x,ξ) =(a ij ) p×p ≥0, here a ij =a ij (x,ξ) (i,j= 1,P ──,α≤ξ≤x≤β). Let φ(x)=(φ 1(x),φ 2(x),φ p(x)) τ. Ψ(x)=(Ψ 1(x),Ψ 2(x),Ψ p(x)) τ. C(x)=(C 1(x),C 2(x),…,C p(x)) τ,if φ(x)≤C(x)+∫ x αA(x,ξ)φ(ξ)dξ, then we have φ(x)≤Ψ(x) (α≤x≤β). Here Ψ(x) is only solution of the integral equation Ψ(x)=C(x)+∫ x αA(x,ξ)Ψ(ξ)dξ. Besides, the relevant conclusion of the above integral inequality is also valid when the “≤"is changed into “<,≥,>”。