ON CONVEX AND CONCAVE SEQUENCES AND THEIR APPLICATIONS.

The aim of this paper is to introduce and to investigate the basic properties of qconvex, 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 qconcave 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. [ABSTRACT FROM AUTHOR]