含函数微分的积分不等式

Integral Inequality Including Function Derivate

利用变分方法构造并证明了含变元微分的积分不等式方法,并用该方法推广了Hilbert不等式和Opial不等式,求出了下述不等式的最优常数:∫Ω∫ΩF(x,y)f(x)g(y)dxdy≤C∫Ωp(x)[Dβ1 f(x)]2 dx∫[Ωp(x)[Dβ2 g(x)]2 dx]1/2,C1∫Ωp(x)[Dαf(x)]2 dx≤∫Ω∫ΩF(x,y)f(x)Dαf(y)dxdy≤C2∫Ωp(x)[Dαf(x)]2 dx,其中F,p为正定函数.

Variational method was used to construct the method to demonstrate integral inequality including function derivate and to generalize Hilbert inequality and Opial inequality.We obtained the optimal constant of the following inequalities∫Ω∫ΩF（x,y）f（x）g（y）dxdy≤C1/2, C1∫Ω p（x）[Dαf（x）]2dx≤∫Ω∫ΩF（x,y）f（x）Dαf（y）dxdy≤C2∫Ω p（x）[Dαf（x）]2dx,where F,p are positive definite functions.