Recently, the authors have shown that Gaussian elimination is stable for complex matrices A=B+iC where both B and C are Hermitian definite matrices. Moreover, the growth factor is less than $3\sqrt 2$ under any diagonal pivoting order. Assume now that B and C, in addition to being (positive) definite, satisfy the inequality $\displaystyle C \le \alpha B, \quad \alpha \ge 0, $ i.e., $ \displaystyle (Cx,x) \le \alpha (Bx,x) \quad \forall x \in {\bf C}^n. $ If α = 0, then A = B is a Hermitian positive definite matrix. It is well-known that, in this case, the growth factor is equal to 1. For α > 0, we establish a bound for the growth factor that has the limit 1 as α → 0.