Some inequalities for generalized convex functions of several variables (Q1180747)

The author defines a two-place function as \(P\)-convex of order \(k\) if the \(i\)-th divided difference with respect to its first variable of the \((k- i)\)-th divided difference with respect to its second variable is nonnegative for all \(i\in\{0,1,\ldots,k\}\). He offers, among others, the following generalization of the Jensen-Steffensen inequality. If \(a\leq x_1\leq\cdots\leq x_n\leq b\), \(c\leq y_1\leq\cdots\leq y_n\leq d\) and \(p_1,\dots,p_n\geq 0\) with \(\sum p_ j>0\) then, for all \(P\)-convex functions \(f: [a,b]\times[c,d]\to\mathbb{R}\), \[ f(\sum p_jx_ j/\sum p_j, \sum p_jy_ j/\sum p_ j)\leq\sum p_j f(x_j,y_j)/\sum p_j. \] \{There are several misprints, such as \(x_k\) instead of \(x_i\) in (1), \(y_m\) instead of \(y_k\) in the next line, \(F\) instead of \(f\) in the line between (11) and (12), etc. The last five lines of p. 86 are completely confused and confusing; anyway the notation \(xy\), which they try to define, is not very fortunate.\}.