Some New Inequalities for the Strictly Monotonic Functions and Positive n-Tuples

Abstract

In this paper we give the generalization of the inequalities
\begin{equation*}
\begin{split}
&n\sum^n_{i=1}a_i^{n} \geq \sum^n_{i=1}a_i^{t} \sum^n_{i=1}a_i^{n-t} \geq \sum^n_{i=1}a_i^{s} \sum^n_{i=1}a_i^{n-s} \geq (\sum^n_{i=1}a_i^\frac{n}{2})^{2} ,\\ & 0 < t < s < \frac{n}{2},\ a_i>0,\forall i \in \{{1,\dots,n}\},
\end{split}
\end{equation*}
(see [3] and [1, p.~377], and the new results for the strictly monotonic functions and positive n-tuples.