Characterization of transfer functions with no local peaks
As far as I know there is no simple characterization of filters with a monotonic magnitude response. I could only come up with a weak time domain characterization of LTI systems with monotonically decreasing amplitude for positive frequencies (cf. Eq. $(6)$ below).

Let $h(t)$ be the real-valued impulse response of an LTI system, and let $H(\omega)=A(\omega)e^{j\phi(\omega)}$ be its Fourier transform. Furthermore, define $A(\omega)$ and $\phi(\omega)$ such that $A(\omega)\ge 0$ is satisfied. Now we require

$$\frac{dA(\omega)}{d\omega}\le 0,\qquad\omega&gt;0\tag{1}$$

Let $r_h(t)$ be the deterministic auto-correlation function of $h(t)$:

$$r_h(t)=\int_{-\infty}^{\infty}h(\tau)h(\tau+t)d\tau\tag{2}$$

Note that the Fourier transform of $r_h(t)$ is given by $A^2(\omega)$. By a basic property of the Fourier transform we have

$$\begin{align}t\,r_h(t)&amp;=\mathcal{F}^{-1}\left\{j\frac{dA^2(\omega)}{d\omega}\right\}\\&amp;=\frac{j}{2\pi}\int_{-\infty}^{\infty}\frac{dA^2(\omega)}{d\omega}e^{j\omega t}d\omega\\&amp;=-\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{dA^2(\omega)}{d\omega}\sin(\omega t)d\omega\\&amp;=-\frac{1}{\pi}\int_{0}^{\infty}\frac{dA^2(\omega)}{d\omega}\sin(\omega t)d\omega\tag{3}\end{align}$$

Note that since $A(\omega)\ge 0$, condition $(1)$ implies

$$\frac{dA^2(\omega)}{d\omega}\le 0,\qquad\omega&gt;0\tag{4}$$

From $(3)$ and $(4)$ we get the following inequality:

$$|t\, r_h(t)|\le -\frac{1}{\pi}\int_{0}^{\infty}\frac{dA^2(\omega)}{d\omega}d\omega=\frac{A^2(0)-A^2(\infty)}{\pi}\tag{5}$$

which finally results in

$$|r_h(t)|\le \frac{A^2(0)-A^2(\infty)}{\pi |t|},\qquad t\neq 0\tag{6}$$

In sum, condition $(1)$ implies that the deterministic auto-correlation of the corresponding impulse response must satisfy inequality $(6)$.