The behavior of special differentiable real-valued functions defined o
n a set of matrices is examined. We call these functions (M; U) functi
ons. They are intrinsically interesting and can provide useful inequal
ities. We give a theorem helping to find their global extrema The sear
ch for these extrema is performed through classical techniques using d
ifferential calculus. We concentrate on (M; U) functions that are impo
rtant in statistics and econometrics. Their global etremizers are expl
icitly given. These functions are useful in the minimization of mean s
quared errors or variances of quadratic forms y'Ay when y has a multiv
ariate normal or, more generally, an elliptically contoured distributi
on. Finally, an illustrative application to the estimation of the cova
riance matrix in a linear regression model is given.