Investigating the achievable rate of noncoherent Gaussian fading  channels leads to the expected value of the log det of certain random matrices. Upper bounds for the expectation are easy to derive from Jensen's inequality. Lower bounds, however, are intriguingly difficult to obtain. In this presentation we show a new inequality for the expectation E[log det (WQ + I)], where Q is a nonnegative definite matrix and W is a diagonal random matrix with identically distributed, but not necessarily independent, nonnegative diagonal entries. A sharp lower bound is obtained by substituting Q by the diagonal matrix of its eigenvalues. From this general result we derive related deterministic inequalities of Muirhead- and Rado-type.