On convex and concave sequences and their applications
Classical Analysis and ODEs
2021-12-21 v1
Abstract
The aim of this paper is to introduce and to investigate the basic properties of -convex, -affine and -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 -concave sequences are the pointwise minima of -affine sequences. As an application, we consider a nonlinear selfmap of the -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 -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.
Cite
@article{arxiv.2112.10197,
title = {On convex and concave sequences and their applications},
author = {Gábor Marcell Molnár and Zsolt Páles},
journal= {arXiv preprint arXiv:2112.10197},
year = {2021}
}