A companion to Jensen-Steffensen's inequality
Suppose that f is a convex function on (a,b). Let \(a<x_ i\leq...\leq x_ n<b\); \(p_ 1,...,p_ n\in R\); \(P_ n=p_ 1+...+p_ n\) and \(A=\sum^{n}_{i=1}p_ ix_ if_+'(x_ i)/\sum^{n}_{i=1}p_ if_+'(x_ i).\) The following companion to Jensen-Steffensen's inequality is proved: If \(0\leq \sum^{K}_{i=1}p_ i\leq P_ n\) (1\(\leq k\leq n-1)\), \(P_ n>0,\sum^{n}_{i=1}p_ if_+'(x_ i)\neq 0\) and \(A\in (a,b)\), then \[ P_ n^{-1}\sum^{n}_{i=1}p_ if(x_ i)\leq f(\sum^{n}_{i=1}p_ ix_ if_+'(x_ i)/\sum^{n}_{i=1}p_ if_+'(x_ i)). \] For a nondecreasing function f and nonnegative \(P_ i\) this implies \textit{M. L. Slater's} result [ibid. 32, 160-166 (1981; Zbl 0521.41011)].
scientific article; zbMATH DE number 3979676