this question is closely related to my other question from yesterday : Interpolation of three positive values at 0, 1 and 2 by a polynomial with non-negative coefficients.
Let $ g $ be a univariate polynomial with real coefficients (actually rational coefficients, but I don't think this will matter) and $ g(0) = 1 $. Is there always another real polynomial $ h $ such that the product $ g h $ satisfies $ (gh)(0) = 1 $ (which, of course implies $ h(0) = 1 $ ) and all coefficients but the absolute one in $ gh $ are non-positive and $ -1 \lt (gh)(1) - 1 \lt 1 $ and $ -2^{r+s} \lt (gh)(2) - 1 \lt 2^{r+s} $ where $ r $ and $ s $ are the degrees of $ g $ and $ h $ ?
Thank you for thinking about it.