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 $f(x)=a_0+a_1x+\cdots + a_mx^m$ 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 $M_*(P,d_1)\geq M_*(P,d_2)$ and $M^*(P,d_1)\geq M^*(P,d_2)$. We give a proof of this conjecture.