A classic Cauchy-Schwarz problem is to show that if $p_1, p_2 \cdots p_k$ are positive real numbers with $p_1 + p_2 \cdots +p_k=1$ then $\sum_{k=1}^n (\frac{1}{p_k} +p_k)^2 \geq n^3 +2n + \frac{1}{n}$ with equality if and only if $p_i=\frac{1}{n}$ for all $i$.
It is also not hard to use Cauchy-Schwartz to show that if $p_1, p_2 \cdots p_k$ are positive real numbers with $p_1 + p_2 \cdots +p_k=1$ then $\sum_{k=1}^n (\frac{1}{p_k} +p_k) \geq n^2 +1$.
Taken together, these suggest the following generalization, which I'm hoping to see a proof or disproof of. If $p_1, p_2 \cdots p_k$ are positive real numbers with $p_1 + p_2 \cdots +p_k=1$
$\sum_{k=1}^n (\frac{1}{p_k} +p_k)^m \geq n(\frac{1}{n} +n)^m$ with equality only when the $p_i$ are all equal.
The general inequality is easy to see for $n=2$, from just looking at the function $f(x)=(\frac{1}{x}+x)^m$, since its derivative is zero at $x=1/2$ and then one can apply the first derivative test.
