On the Modes of Polynomials Derived from Nondecreasing Sequences

Wang and Yeh proved that if P (x) is a polynomial with nonnegative and nondecreasing coefficients, then P (x + d) is unimodal for any d > 0. A mode of a unimodal polynomial f(x) = a0 + a1x + · · · + amx m is an index k such that ak is the maximum coefficient. Suppose that M∗(P, d) is the smallest mode of P (x + d), and M(P, d) the greatest mode. Wang and Yeh conjectured that if d2 > d1 > 0, then M∗(P, d1) ≥ M∗(P, d2) and M (P, d1) ≥ M (P, d2). We give a proof of this conjecture.