On global bounds for generalized Jensen's inequality (Q1947136)

Let \(f:[a,b]\rightarrow \mathbb{R}\) be a convex function. Let \(x_i\in [a,b]\), \(p_i>0\), \(i=1,2,\dots ,n\) be such that \(\sum_{i=1}^np_i=1.\) The author proves that \[ \sum_{i=1}^np_if(x_i)-f\left (\sum_{i=1}^n p_ix_i\right )\leq \max_p\{pf(a)+qf(b)-f(pa+qb)\}, \] with \(p,q\geq 0\) such that \(p+q=1\). The author applies this result to obtain some inequalities for integrals and means.