Local comparisons of homological and homotopical mixed Hodge polynomials

For a simply connected complex algebraic variey $X$, by the mixed Hodge structures $(W_{\bullet },F^{\bullet })$ and $(\tilde W_{\bullet}, \tilde F^{\bullet})$ofthehomologygroup$H_{*}(X;\mathbb Q)$ and the homotopy groups ${\pi }_{\ast }(X)\otimes \mathbb{Q}$ respectively, we have the following mixed Hodge polynomials $$M{H}_X(t,u,v):=\sum _{k,p,q}\dim (G{r}_{F^{\bullet }}^{p}G{r}_{p+q}^{W_{\bullet }}{H}_k(X;\mathbb{C}))t^k{u}^{-p}{v}^{-q},$$ $$M{H}_X^{\pi }(t,u,v):=\sum _{k,p,q}\dim (G{r}_{{\tilde{F}}^{\bullet }}^{p}G{r}_{p+q}^{{\tilde{W}}_{\bullet }}({\pi }_k(X)\otimes \mathbb{C}))t^k{u}^{-p}{v}^{-q},$$ which are respectively called \emph{the homological mixed Hodge polynomial} and \emph{the homotopical mixed Hodge polynomial}. In this paper we discuss some inequalities concerning these two mixed Hodge polynomials.