Derivates, approximate derivates and porosity derivates of \(n\)-convex functions (Q1852457)

Let \(x_0,\dots, x_n\) be \((n+1)\) distinct points from \([a,b]\). The \(n\)th divided difference of \(f: [a,b]\to\mathbb{R}\) at these \((n+1)\) points is defined by \(V_n(f; x_k)= \sum^n_{k=0} f(x_k)/w'(x_k)\), where \(w(x)= w_n(x,x_k)= \prod^n_{k=0} (x-x_k)\). A function \(f\) is said to be \(n\)-convex on \([a,b]\) if for all choices of \((n+1)\) distinct points \(x_0,\dots, x_n\) in \([a,b]\), \(V_n(f;x_k)\geq 0\). The authors show that for an \(n\)-convex function \(f\) the four \(n\)th-order Peano derivates of \(f\) are, resp., equal to the corresponding \(n\)th-order approximate Peano derivates and the porosity Peano derivates of \(f\). It is further shown that the same result holds for the de la Vallée Poussin derivates, and the symmetric and unsymmetric Riemann derivates. This result generalizes, in some sense, a known fact that for a monotone function \(f\) the four Dini derivates of \(f\) are equal to the corresponding approximate derivates of \(f\).