\({\lambda{}}(n)\)-convex functions (Q1191828)

\{It is not easy to find out what this paper is about. The author prefers to define by giving very special examples, sometimes followed by exact definitions, sometimes not, and sometimes the explanation of symbols is completely missing. On the ninth page of the paper an Introduction (sic) starts, mostly historical, rather interesting but not very enlightening concerning the contents of the present paper. So one turns to the abstract which says verbatim: ``The goal of this research is to characterize the \(\lambda(n)\)-convex functions in terms of determinants and divided differences. The results of this paper do not appear in any published mathematical literature.''\} The following may give some information of the basic definition. Let \(\lambda_ 1,\ldots,\lambda_ k\) be positive integers, \(n=\lambda_ 1+\cdots+\lambda_ k\), \(\lambda(n):=(\lambda_ 1,\ldots,\lambda_ k)\), and \(I\subseteq \mathbb{R}\) an interval containing \(x_ 1<x_ 2<\cdots<x_ k\). Let \(f:I \to \mathbb{R}\) be \(\lambda_ j-1\) times differentiable at \(x_ j\) and \(p_ n(x)\) the Hermite interpolation polynomial, the values and up to \(\lambda_ j-1\)-st (not \(\lambda_{j-1}\) as misprinted in the paper) derivatives of which agree with those of \(f\) at \(x_ j\) \((j=1,\ldots,k)\). If \(f(x)-p_ n(x)\) keeps the appropriate sign on \(]x_{j-1},x_ j[\) and changes it at \(x_ j\) for odd \(\lambda_ j\) but keeps it for even \(\lambda_ j\) \((j=1,\ldots,k)\) then \(f\) is \(\lambda(n)\)-convex on \(I\). Now the paper indeed offers necessary and sufficient conditions for \(\lambda(n)\)-convexity in terms of Vandermonde determinants and of divided differences of \(n\)-th order (really of divided differences of derivations if there are \(\lambda_ j\) greater than 1).