I have a matrix problem that I need help with.

Let H=aA+bB, where a+b=1,a>0,b>0,and A,B are matrices having non-negative real part eigenvalues. In addition, A+B has positive real part eigenvalues. I also know that all the eigenvalues of H have positive real parts. I want to get a lower bound on the minimum real part of the eigenvalues of H in terms of a,b and real parts of the eigenvalues of A+B.

Thanks in advance.