积分中值定理中间点比较及有关平均不等式

中值定理中间点是区间端点的平均.设f (x)、g(x)在同一区间[a,b]内严格单调并可积,p(x)、q(x)恒正可积,按积分中值定理各有唯一的中间点ξf ,p(a,b)和ξg,q(a,b) .当f递增(减)且f (g- 1)凸(凹)时,有ξg,p(a,b) <ξf,p(a,b) ;当p(x)q(x) 递增(减)且q(x) ∫bap(x) dx >( <) 0时,有ξf,q(a,b) <ξf ,p(a,b) .由此可证明和发现一系列有关平均的不等式.

Comparison of Interior Points for Mean Value Theorem for Integrals and Inequalities for Mean Involved

The interior point for mean value theorem is a mean between endpoints of an interval. Let f(x), g(x) be strictly monotone and integrable, p(x), q(x) be always positive and integrable over the same interval [a, b], according to mean value theorem for integrals, there is sole mean value ξ f,p (a,b) and ξ g,q (a,b) respectively. when f is increasing (decreasing) and f(g -1 ) is convex (concave), we have ξ g,p (a,b)<ξ f,p (a,b); when p(x)q(x) is increasing (decreasing) and q(x)∫ b ap(x)dx>(<)0, we have ξ f,q (a, b)<ξ f,p (a,b). From it we can prove and find out a series of inequalities for means.