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) = a(0) + a(1)x + ... + a(m)x(m) is an index k such that a(k) 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 d(2) > d(1) > 0, then M*(P, d(1)) >= M*(P, d(2)) and M*(P, d(1)) >= M*(P, d(2)) . We give a proof of this conjecture.