Improvement of Jensen-Steffensen's inequality for superquadratic functions
Jensen-Steffensen's inequality states that if \(f : I \to \mathbb{R}\) is a convex function, then \[ f \left ({1\over {A_n}} \sum_{i=1}^n a_i x_i \right) \leq {1\over {A_n}} \sum_{i=1}^n a_i f( x_i ) , \] where \(I\) is an interval in \(\mathbb{R}\), \( ( x_1,x_2,\dots,x_n)\) is any monotonic \(n\)-tuple in \(I^n\), and \((a_1, a_2, \dots,a_n )\) is a real \(n\)-tuple that satisfies \[ 0 \leq A_j \leq A_n, \quad A_n > 0 ,\quad A_j = \sum_{i=1}^j a_i , \quad \bar{A_j} = \sum_{i=j}^n a_i , \quad j=1,2,\dots, n. \] In this paper, the authors prove some inequalities for superquadratic functions analog to Jensen-Steffensen's inequality for convex functions. For superquadratic functions which are convex, the authors prove some improvements and extensions of Jensen-Steffensen's inequality.
scientific article; zbMATH DE number 5702406