I'm interested in getting bounds on the following summation for fixed discrete probability distribution $p_i$ over $d$ outcomes with $p_i>0$ and arbitrary $k>1$
$$f(k)=\sum_{i=1}^d p_i (1-p_i)^k$$
I can get upper bounds by applying Jensen's inequality to subsets of numbers, which gives me a family of cheap upper bounds.
$$\sum_i^s a_i^k\le \left(\sum_i^s a_i\right)^k$$
Is there a good family of cheap lower bounds? Especially for the case when $k\approx d$ and $d$ is large.
Motivation: I'm using root finder to solve for $\sum_i^d p_i (1-p_i)^k=b$ for this application which is too expensive because $d$ is large, so I want to speed it up by finding bounds first
The lower bound is unfortunately $0$. Indeed, let $p_1=p_2=\dots=p_{d-1}=0$, and $p_d=1$. Of course, you require them to be all positive, but you can make them very very small, and the last one to be close to one.
By the way, there's a nice upper bound using the fact that $$ \sup_{p\in[0,1]} p(1-p)^k=\frac{k^k}{(k+1)^{k+1}} $$ so $$ \sum_{i=1}^d p_i(1-p_i)^d\le \frac{k^k d}{(k+1)^{k+1}}. $$