For
$$m(\theta) = b_{0} + b_{1}\theta + b_{2}\theta^2 + \cdots +
b_{2k+1}\theta^{2k+1}$$
to be a monotonic function, a necessary and sufficient condition is that its
first derivative,
$$p(\theta) = a_{0} + a_{1}\theta + ... + a_{2k}\theta^{2k},$$
is nonnegative at all theta. Here, let
$$b_{0} = \xi$$
be the constant of integration and
$$b_{s} = a_{s-1}/s$$
for \(s = 1, 2, ..., 2k+1\).
Notice that \(p(\theta)\) is a polynomial function of degree \(2k\).
A nonnegative polynomial of an even degree can be re-expressed as the
product of k quadratic functions.
If \(k \geq 1\):
$$p(\theta) = \exp{\omega} \Pi_{s=1}^{k}[1 - 2\alpha_{s}\theta +
(\alpha_{s}^2+ \exp(\tau_{s}))\theta^2]$$
If \(k = 0\):
$$p(\theta) = 0.$$