关于Iyengar不等式

On Iyengar Inequaltiy

利用函数f(x)在积分区间 [a ,b]端点的函数值及各阶导数值 ,对函数f(x)在 [a ,b]上的定积分进行估计 ,进而得到若干积分不等式 .主要结果如下 :若函数f(x)是 [a ,b]上n +1次可微函数 ,且 |f(n + 1) (x) |≤M(M >0 ) ,则∫baf(x)dx-∑nk=0(b-a) k+ 12 k+ 1(k+ 1 ) ! [f(k) (a) + (-1 ) kf(k) (b) ]≤ 12 n+ 1(n+ 2 ) ! M(b-a) n+

By using the values of function f(x) and its derivatives in different orders at the end point of integral interval [a,b] to estimate the definite integral of f(x) on [a,b],several integral inequalities are obtained.The main result is as follows:If f(x) is a differentiable function of n+1 order on [a,b],and |f (n+1) (x)|≤M(M>0),then∫ b af(x) d x-∑nk=0(b-a) k+1 2 k+1 (k+1)![f (k) (a)+(-1) kf (k) (b)]≤12 n+1 (n+2)!M(b-a) n+2 .