An extremal problem for positive defintie matrices

A problem studied by Flanders (1975) is minimize the function f(R)=tr(SR+TR^-1) over the set of positive definite matrices R, where S and T are positive semi-definite matrices. Alternative proofs that may have some intrinsic interest are provided. The proofs explicitly yield the infimum of f(R). One proof is based on a convexity argument and the other on a sequence of reductions to a univariate problem.