We consider t-input r-output Rayleigh fading channels with transmit-sided correlation, where the receiver knows the channel realizations, and the transmitter only knows the channel statistics. Using Lagrange duality, we develop an easily computable, tight upper bound on the loss in information rate due to the use of any given input covariance for this channel. This bound is applied to two simple transmission strategies. The first strategy is a reduced-rank uniform allocation, in which independent, equal power Gaussian symbols are transmitted on the at strongest eigenvectors of the transmit covariance matrix, where 0 les alpha les 1 is chosen to optimize the resulting information rate. The second strategy is water-filling on the eigenvalues of the transmit covariance matrix. The upper bound on loss shows these strategies are nearly optimal for a wide range of signal to noise ratios and correlation scenarios