On some inequalities for \(\nabla\)-convex sequences of higher order (Q1061325)

Let \(P=\| p_{n,i}\|\), \(i=0,...,n\); \(n=0,1,..\). be a triangular matrix which transforms the sequence \(a=(a_ n)\), \(n=0,1,..\). in the following way \(A_ n(a)=\sum^{n}_{k=0}p_{n,n-k^ ak},\) \(n=0,1,... \). In this work, necessary and sufficient conditions for P are given so that \(\nabla^ ma_ n\geq 0\) implies \(\nabla^ sA_ n(a)\geq 0\) where \(\nabla^ k\) is the backward difference of order \(k=1,2,... \).