We say that an arithmetic function $f:\mathbb{Z}^+\to\mathbb{C}$ is Möbius-monotone if $\forall n\geq1:(\mu*f)(n)\geq0$ (where $*$ denotes de Dirichlet convolution), i.e. if there exists a non-negative arithmetic function $g:\mathbb{Z}^+\to\mathbb{C}$ such that $f(n) = \sum_{d\vert n}g(d)$.
The problem now is: is there a 'nice' characterization of Möbius-monotonicity? 'Nice' in the sense that it doesn't require to explicitly compute the Möbius inverse $g=\mu*f$.
For instance, we know that if $f$ is Möbius-monotone then $\forall n\geq1:f(n)=(1*g)(n)\geq0$ and if $n,k\geq1$ then $f(kn)\geq f(n)$ $$f(kn)=\sum_{d\vert kn}g(d)=\sum_{d\vert n}g(d)+\sum\limits_{d\vert kn\\ d\not\vert n}g(d)\geq \sum_{d\vert n}g(d)=f(n)$$ Hence, a necesary condition for Möbius-monotonicity would be to satisfy $\forall n,k\geq1: f(kn)\geq f(n)\geq0$. But is this condition sufficient as well? Notice also that $\forall n,k\geq1:f(kn)\geq f(n)$ can be expressed as $\forall n,d\geq1:d\vert n\Rightarrow f(d)\leq f(n)$.
$\bf Edit:$ Now I know that the condition $\forall n,k\geq1: f(kn)\geq f(n)\geq0$ is not sufficient for Möbius-monotonicity. If we consider the function $f(n) := \max\{p\in\mathbb{P}\cup\{1\}:p\vert n\}$ we see that $$f(kn)=\max\{p\in\mathbb{P}\cup\{1\}:p\vert kn\}\geq \max\{p\in\mathbb{P}\cup\{1\}:p\vert n\} = f(n)\geq0$$ $$\text{but } (\mu*f)(6) = f(6)-f(3)-f(2)+f(1)=3-3-2+1=-1<0$$ therefore a counterexample.