Let $a_1 \ge a_2 \ge \cdots \ge a_n \ge 1$ or $0 \le a_1 \le a_2 \le \cdots \le a_n \le 1$ and $\alpha_1 \ge \alpha_2 \ge \cdots \ge \alpha_n \ge 1$ then
$$\left(\sum_{i=1}^{n}{\alpha_i} \right)\left(\sum_{i=1}^{n}{a_i^{\alpha_i}} \right) \ge \sum_{i=1}^{n}{\alpha_i} \left(\sum_{j=1}^{n}{a_i^{\alpha_j}} \right)$$
Equality holds if on ly if $a_1=a_2=\cdots=a_n$ or $\alpha_1=\alpha_2=\cdots=\alpha_n$
Special case: $n=2, \alpha_2=1$, $\alpha_1=\alpha \ge 1$ and $a_2=1, a_1 = a >0$ we have:
$$(\alpha+1)(a^\alpha+1^1) \ge \alpha(a^\alpha+a)+1(1^\alpha+1^1)$$
$$ \Leftrightarrow a^\alpha+\alpha \ge \alpha a+1 $$
This is Bernoulli's inequality
Aplication: Diophantine equation
$$yx^x+xy^y=x^{y+1}+y^{x+1}$$
has only positive integer solution $x=y$
History: I found and prove the inequality above more ten years ago. But I didn't publish. I think the inequality is essential of exponentiation function. So I pose to here and hope that some one like this and found interesting application.
My question: What does interesting application of the inequality? May You research detail?