I'm considering a function $f(x)$ defined over some range $[a,b]$ that is infinitely differentiable with the conditions that $f'(x)\geq 0$, $f''(x)\leq 0$, $f'''(x)\geq 0$, ..., i.e.,
$$(-1)^{n-1}f^{(n)}(x)\geq 0, \forall\,n\in\mathbb{N}^*, x\in[a,b].$$
For example, $f(x)=\ln x\,$ for $\,0<a<b\,$ satisfies the above conditions. An interesting property of the class $\,\mathcal{C}\,$ of such functions $\,f(x)\,$ that satisfy the above conditions is that if $\,f(x)\in\mathcal{C}\,$, so does $-f'(x)$. Is there a name for the class $\,\mathcal{C}\,$ of such functions? Also, if I'm fitting some noisy data points $\,(x_i,y_i)\,$ sampled on equal intervals of $\,\{x_i\}\,$ coordinates using least squares errors to a function $\,y=f(x)\in\mathcal{C}$, what is an efficient way to enforce the constraint $f(x)\in\mathcal{C}\,$?