What methods can be used to find upper and lower bounds for a sequence of non-zero complex numbers b in terms of each other given they obey the following conditions: $$\sum_{n=1}^m b_n=0\qquad \text{ and }\qquad \prod_{n=1}^m b_n=1\qquad \text{ and }\qquad \sum_{n=1}^m 1/b_n=0$$
The only solution I have found to this sequence B was $$b_n=-(-1)^m(c_m)^n$$ where $c_m$ is the mth root of unity. Are there others or at least bounds on their solutions?
Choose them arbitrarily for $n=1$ to $m-2$.
Then want $0 =\sum_{n=1}^m b_n =\sum_{n=1}^{m-2} b_n+b_{m-1}+b_m $ and $1 =\prod_{n=1}^m b_n =\prod_{n=1}^{m-2} b_n\cdot b_{m-1}\cdot b_m $.
Letting $u=\sum_{n=1}^{m-2} b_n, v=\prod_{n=1}^{m-2} b_n, x=b_{m-1}, y=b_m $, this becomes $0=u+x+y, 1=vxy $.
Solve this for $x$ and $y$ in terms of $u$ and $v$.
As to bounds, I don't know.