Inheriting Inequality

Problem

Inheriting Inequality

Solution

The answer is, Yes, and follows from Karamata's theorem. The sequences can be so rearranged as to satisfy Karamata's majorization requirement. Since the function $f(x)=x^p$ is convex and strictly increasing for $p\gt 1$ Karamata's theorem applies to establish $\displaystyle \sum_{k=1}^na_k^p\ge\sum_{k=1}^nb_k^p.$

Acknowledgment

The question arose following N.N. Taleb's similar inquiry with $p\lt 1.$

 

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