A comparison between homogeneous and heterogeneous portfolios

We compare two portfolios: in the heterogeneous portfolio the individual risks are independent but not identically distributed. In the homogeneous portfolio the risks are independent and identically distributed. We compare the heterogeneous portfolio with two types of homogeneous portfolios. First, we assume that the distribution of each risk in the homogeneous portfolio is a mixture with equal weights of the risks in the heterogeneous portfolio, and get an upper bound for the heterogeneous portfolio. To get a lower bound we assume that the risks in the homogeneous portfolio are the average of the individual risks in the heterogeneous portfolio.