Integrally closed ideals of reduction number three

In a Cohen-Macaulay local ring $(A,\mathfrak{m})$, we study the Hilbert function of an integrally closed $\mathfrak{m}$-primary ideal $I$ whose reduction number is three. With a mild assumption we give an inequality $\ell_A(A/I) \ge $\mathrm{e}$_0(I) - $\mathrm{e}$_1(I) + \dfrac{$\mathrm{e}$_2(I) + \ell_A(I^2/QI)}{2}$,where$$\mathrm{e}$_i(I)$denotesthe$i$th Hilbert coefficients and $Q$ denotes a minimal reduction of $I$. The inequality is located between inequalities of Itoh and Elias-Valla. Furthermore our inequality becomes an equality if and only if the depth of the associated graded ring of $I$ is larger than or equal to $\dim A-1$. We also study the Cohen-Macaulayness of the associated graded rings of determinantal rings.