Cyclic inequalities involving Cater cyclic function

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: $$\sum_{i=1}^{n} {a_i^{{a_{n+1-i}}}}\leq\sum_{\text{cyc}:~n}^{1\leq i\leq n} {a_i^{{a_{i + 1}}}}\leq \sum_{i=1}^{n} {a_i^{{a_i}}},~\forall n\geq2.$$