  Let $\mathcal{F}$ be a holomorphic foliation at $p\in \mathbb{C}^2$, and $B$
be a separatrix of $\mathcal{F}$. We prove the following Dimca-Greuel type
inequality $\frac{\mu_p(\mathcal{F},B)}{\tau_p(\mathcal{F},B)}<4/3$, where
$\mu_p(\mathcal{F},B)$ is the multiplicity of $\mathcal{F}$ along $B$ and
$\tau_p(\mathcal{F},B)$ is the dimension of the quotient of $\mathbb{C}[[x,y]]$
by the ideal generated by the components of any $1$-form defining $\mathcal{F}$
and any equation of $B$. As a consequence, we provide a new proof of the
$\frac{4}{3}$-Dimca-Greuel's conjecture for singularities of irreducible plane
curve germs, with foliations ingredients, that differs from those given by
Alberich-Carrami\~nana, Almir\'on, Blanco, Melle-Hern\'andez and
Genzmer-Hernandes but it is in line with the idea developed by Wang.
