An upper bound for the GSV-index of a foliation

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 $3\mu_p(\mathcal{F},B)-4\tau_p(\mathcal{F},B)+GSV_p(\mathcal{F},B)\leq 0$, where $\mu_p(\mathcal{F},B)$ is the multiplicity of $\mathcal{F}$ along $B$, $\tau_p(\mathcal{F},B)$ 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 $B$, and $GSV_p(\mathcal{F},B)$ is the \textit{G\'omez-Mont-Seade-Verjovsky index} of the foliation $\mathcal{F}$ with respect to $B$. As a consequence, we provide a new proof of the $\frac{4}{3}$-Dimca-Greuel 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.