On the Jensen-Steffensen inequality for generalized convex functions (Q950292)

The main results of the paper concern two classes of functions: \(P\)-convex functions and functions with nondecreasing increments. Given two real intervals \(I\) and \(J\), a real valued function \(\varphi\) defined on \(I\times J\) is called \(P\)-convex if the inequalities \[ [x_0,x_1,x_2][y_0]\varphi\geq0,\qquad [x_0,x_1][y_0,y_1]\varphi\geq0,\qquad [x_0][y_0,y_1,y_2]\varphi\geq0 \] hold for all \(x_0,x_1,x_2\in I\), \(y_0,y_1,y_2\in J\) with \(x_0<x_1<x_2\) and \(y_0<y_1<y_2\). (Here \([u_0,\dots,u_k]\) denotes the \(k\)-th order divided difference operator with respect to the pairwise distinct nodes \(u_0,\dots,u_k\)). Given an \(n\)-dimensional rectangle \(I\), a real valued function \(f\) on \(I\) is said to have nondecreasing increments if \[ f(x+h)-f(x)\leq f(y+h)-f(y) \] holds whenever \(0\leq h\), \(x\leq y\), \(x,y+h\in I\) (where ``\(\leq\)'' means the standard componentwise ordering of \(n\)-dimensional vectors). Motivated by A.\ Mercer's generalization of the Jensen inequality, the authors prove a Jensen-Steffensen type inequality for \(P\)-convex functions and also for functions with nondecreasing increments. As applications, generalizations of Čebyšev and Hölder inequalities and also of inequalities related to generalized quasi-arithmetic means are obtained.