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, qaffine 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.