Infinite sequences and their h-type indices

Abstract

Starting from the notion of h-type indices for infinite sequences we investigate if these indices satisfy natural inequalities related to the arithmetic, the geometric and the harmonic mean. If f denotes an h-type index, such as the h- or the g-index, then we investigate inequalities such as min(f(X),f(Y)) <= f((X + Y)/2) < max(f(X), f(Y)). We further investigate if: f(min(X,Y)) = min(f(X),f(Y)) and if f(max(X,Y)) = max(f(X),f(Y)). It is shown that the h-index satisfies all the equalities and inequalities we investigate but the g-index does not always, while it is always possible to find a counterexample involving the R-index. This shows that the h-index enjoys a number of interesting mathematical properties as an operator in the partially ordered positive cone (R+)(infinity) of all infinite sequences with non-negative real values. In a second part we consider decreasing vectors X and Y with components at most at distanced. Denoting by D the constant sequence (d,d,d, ...) and by Y-D the vector (max(y(r) - d), 0)(r), we prove that under certain natural conditions, the double inequality h(Y-D) <= h(X) <= h(Y + D) holds. (C) 2019 Elsevier Ltd. All rights reserved.