Upper bound for the sum of non-integer powers

Question

Let $a_1, a_2, \ldots, a_k$ be a positive integers such that $a_1 + a_2 + \cdots + a_k = K$. Is it possible to find an upper bound such that $$a_1^p + a_2^p + \cdots+ a_k^p \le f(K)$$ where $0 < p < 1$, and $f$ is some function? It is easy to see that $a_1^p + a_2^p + \cdots + a_k^p \le K$, but can I get anything better than this? I would appreciate any ideas. Thank you!

Answer 1

For fixed $K$ and arbitrary $k$ there is no better bound because we can choose $k=K$ and all $a_j$ equal to one.

For fixed $K$ and fixed $k$ we can apply Jensen's inequality to the concave function $t \mapsto t^p$. This gives $$ \left( \frac K k \right)^p = \left( \frac{a_1 + \ldots + a_k}{k} \right)^p \ge \frac{a_1^p + \ldots + a_k^p}{k} $$ so that $$ a_1^p + \ldots + a_k^p \le k^{1-p} K^p \, . $$ If $K$ is an integral multiple of $k$ then the bound is sharp, as can be seen by choosing all $a_j$ equal to $K/k$.