Approximations of matrix-valued functions of the form $W^Tf(A)W$, where $A \in {\mathbb R}^{m\times m}$ is symmetric, $W\in{\mathbb R}^{m\times k}$, with $m$ large and $ k\ll m$, has orthonormal columns, and $f$ is a function, can be computed by applying a few steps of the symmetric block Lanczos method to $A$ with initial block-vector $W\in{\mathbb R}^{m\times k}$. Golub and Meurant have shown that the approximants obtained in this manner may be considered block Gauss quadrature rules associated with a matrix-valued measure. This paper generalizes anti-Gauss quadrature rules, introduced by Laurie for real-valued measures, to matrix-valued measures, and shows that under suitable conditions pairs of block Gauss and block anti-Gauss rules provide upper and lower bounds for the entries of the desired matrix-valued function. Extensions to matrix-valued functions of the form $W^Tf(A)V$, where $A\in{\mathbb R}^{m\times m}$ may be nonsymmetric, and the matrices $V,W\in{\mathbb R}^{m\times k}$ satisfy $V^TW=I_k$ ...