05A20, 05E05, 15A15, 15A42, 15A45, 26D05

We present in this work a new and shorter proof of the generalized -Newton inequalities for elementary symmetric functions defined on a self-conjugate set which lies essentially in the open right half-plane. We also point out some interesting consequences of the generalized -Newton inequalities. In particular, we establish an improved complex version of the arithmetic mean-geometric mean inequality along with the corresponding determinant-trace inequality for positive stable matrices.;