Cyclic inequalities involving Cater cyclic function

TL;DR

This paper establishes new cyclic inequalities involving the Cater cyclic function using mathematical analysis, induction, and dimension reduction methods under certain hypotheses.

Contribution

It introduces novel cyclic inequalities related to the Cater cyclic function, expanding the theoretical understanding of such inequalities.

Findings

Proved the inequalities under proper hypotheses.

Established bounds involving the Cater cyclic function.

Applied mathematical induction and dimension reduction methods.

Abstract

By means of the mathematical analysis theory, inequality theory, mathematical induction and the dimension reduction method, under the proper hypotheses, we establish the following cyclic inequalities: \[\sum_{i=1}^{n} {a_i^{{a_{n+1-i}}}}\leq\sum_{\text{cyc}:~n}^{1\leq i\leq n} {a_i^{{a_{i + 1}}}}\leq \sum_{i=1}^{n} {a_i^{{a_i}}},~\forall n\geq2.\]