Suppose f(z) = z^n - k [ z^(n-1) + ... + z + 1 ] where n is a positive integer and k is a real constant such that nk<1. I have shown that a root of this polynomial must satisy |z|<1, but I want a slightly better bound such as 1-k. This seems plausible from computational results but is difficult to prove. I am trying to use Rouche's theorem to do this but finding an appropriate bounding function is difficult. Is there any other result about holomorphic functions that may help?