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 $f(x)=a_0+a_{1}x+\cdots +a_m{x}^m$ is an index $k$ such that $a_k$ is the maximum coefficient. Suppose that $M_{\ast }(P,d)$ is the smallest mode of $P(x+d)$, and $M^{\ast }(P,d)$ the greatest mode. Wang and Yeh conjectured that if $d_2>d_1>0$, then $M_{\ast }(P,d_1)\ge M_{\ast }(P,d_2)$ and $M^{\ast }(P,d_1)\ge M^{\ast }(P,d_2)$. We give a proof of this conjecture.