  In this paper we introduce some {\it variation functions} associated to the
rank of the Infinitesimal Variations of Hodge Structure for a family of smooth
projective complex curves. We give some bounds and inequalities and, in
particular, we prove that if $X$ is a smooth plane curve $X$, then there exists
a first order deformation $\xi\in H^1(T_X)$ which deforms $X$ as plane curve,
such that $\xi\cdot:H^0(\omega_X)\to H^1(\mathcal{O}_{X})$ is an isomorphism.
We also generalize the notions of variation functions to the higher dimensional
case and we analyze the link between IVHS and the Weak and Strong Lefschetz
properties of the Jacobian ring of a smooth hypersurface.
