Unimodality and Log-Concavity of Polynomials

A polynomial is unimodal if its sequence of coefficients are increasing up to an index, and then are decreasing after that index. A polynomial is logconcave if the sequence of the logarithms of the coefficients is concave. We prove that if P (x) is a polynomial with nonnegative non-decreasing coefficients then P (x+z) is unimodal for any natural z. Furthermore, we prove that if P (x) is a log-concave polynomial with nonnegative coefficients and no internal zeroes, then for any natural number n, P (x+n) is log-concave. Unimodal polynomials whose coefficients satisfy certain criteria are shown to be log-concave. An open problem in proving a particular polynomial is log-concave is discussed. We prove that if a unimodal polynomial Q(x) satisfies certain conditions, then Q(x− 1) has nonnegative nondecreasing coefficients.