On convex and concave sequences and their applications

Abstract

The aim of this paper is to introduce and to investigate the basic properties of $q$-convex, $q$-affine and $q$-concave sequences and to establish their surprising connection to Chebyshev polynomials of the first and of the second kind. One of the main results shows that $q$-concave sequences are the pointwise minima of $q$-affine sequences. As an application, we consider a nonlinear selfmap of the $n$-dimensional space and prove that it has a unique fixed point. For the proof of this result, we introduce a new norm on the space in terms of a $q$-concave sequence and show that the nonlinear operator becomes a contraction with respect to this norm, and hence, the Banach Fixed Point theorem can be applied.