is the product of a nonnegative monotone increasing and a nonnegative monotone decreasing sequence unimodal?

Question

If there are two sequences

$a_n$ which is a monotonically increasing strictly positive sequence and $b_n$ which is monotonically decreasing strictly positive sequence

is their product

$h_n = a_n b_n$

necessarily a (positive) unimodal sequence ?

Answer 1

Let $b_n=\frac1n$ for $n\in\Bbb Z^+$. Let

$$a_n=\begin{cases} \frac{3n+1}2,&\text{if }n\text{ is odd}\\ \frac{3n}2,&\text{if }n\text{ is even}\;. \end{cases}$$

Then $h_{2k}=\frac32$ for each $k\in\Bbb Z^+$, and

$$h_{2k+1}=\frac{3(2k+1)+1}{2(2k+1)}=\frac{3k+2}{2k+1}>\frac32$$

for $k\in\Bbb N$. Thus,

$$\langle h_n:n\in\Bbb Z^+\rangle=\left\langle 2,\frac32,\frac53,\frac32,\frac85,\frac32,\frac{11}7,\frac32,\ldots\right\rangle\;,$$

oscillating up and down. With a bit of tinkering you can get just about any behavior.