A Nondecreasing Differentiable Majorant with Vanishing Derivative at Zero for a Little-o Modulus

In the setting of Single-Variable Calculus on an Interval, write $[0,\infty)=\{s\in\mathbb{R}:0\le s\}$, and let $[0,r]$ be the closed interval for a real $r>0$. Let $m:[0,r]\to[0,\infty)$ be a function such that $m(0)=0$, there is a real $B$ with $m(s)\le B$ for every $s\in[0,r]$, and for every real $\eta>0$ there is a real $r'>0$ with $m(s)\le\eta s$ for every $s\in[0,r]$ with $s<r'$.

There is a function $h:[0,\infty)\to[0,\infty)$ with the following four properties, where $\bar{h}:\mathbb{R}\to\mathbb{R}$ is given by $\bar{h}(s)=h(s)$ for $s\ge0$ and $\bar{h}(s)=0$ for $s<0$.

(Monotone)¶ $h$ is nondecreasing on $[0,\infty)$ and $h(0)=0$.

(Differentiable)¶ $\bar{h}$ is differentiable at every point of the interval $I=\mathbb{R}$, and we write $h'(s)=\bar{h}'(s)$ for $s\ge0$.

(Derivative)¶ $h'(0)=0$, there is a real $K$ with $|h'(s)|\le K$ for every $s\ge0$, and $h'(s)\to0$ as $s\to0$: for every real $\eta>0$ there is a real $r''>0$ with $|h'(s)|<\eta$ for every $s$ with $0\le s<r''$.

(Majorant)¶ $h(s)\ge m(s)$ for every $s\in[0,r]$.