On the Jensen functional and superquadraticity (Q304014)

Let \(f\) be a real function defined on an interval \(I\), \(\mathbf{x}=(x_1,\dots,x_n)\in I^n, \;\mathbf{p}=(p_1,\dots,p_n)\in (0,1)^n\) with \(\sum_{i=1}^n p_i=1\). The \textit{Jensen functional} is defined by \[ \mathcal{J}(f,\mathbf{p},\mathbf{x})=\sum_{i=1}^np_if(x_i)-f\left(\sum_{i=1}^np_ix_i\right) \] and the \textit{Chebyshev functional} is given by \[ \mathcal{T}(f,\mathbf{p},\mathbf{x})=\sum_{i=1}^np_i\left(x_i-\sum_{i=1}^np_ix_i\right)f(x_i). \] If \(I\) is an interval \([0,a]\) or \([0,\infty)\), then \(f\) is \textit{superquadratic} if for each \(x\) there exists a real number \(C(x)\) such that for all \(y\in I\) \[ f(y)-f(x)\geq f(|y-x|)+C(x)(y-x). \] The authors generalize the definitions of the above functionals and give new upper and lower bounds for them.