Nested unimodality

Let P (x) be a unimodal polynomial of degree m with nonnegative coefficients and a mode n for nonnegative integers n m. We study the unimodality of P (x+ z) for real numbers z = 1 or z 2 and show that: if z = 1, P (x + z) is unimodal provided that m − n 4; if z 2, then P (x+ z) is unimodal provided that m− n 2z + 1; and we also show that the given conditions are best possible. Additionally, we explore the location of modes of P (x+ z), and show P (x+ z) has a mode m−z z+1 or m−z z+1 − 1 or m−z z+1 − 2, which are reachable.