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 $3{\mu }_p(\mathcal{F},B)-4{\tau }_p(\mathcal{F},B)+GS{V}_p(\mathcal{F},B)\le 0$, 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 $GS{V}_p(\mathcal{F},B)$ is the $\mathit{\text{G}}\acute{\mathit{\text{o}}}\mathit{\text{mez-Mont-Seade-Verjovsky}}\mathit{\text{index}}$ 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-Carramiñana, Almirón, Blanco, Melle-Hernández and Genzmer-Hernandes, but it is in line with the idea developed by Wang.