  A generalization of the method developed by Adam, Psarrakos and Tsatsomeros
to find inequalities for the eigenvalues of a complex matrix A using knowledge
of the largest eigenvalues of its Hermitian part H(A) is presented. The
numerical range or field of values of A can be constructed as the intersection
of half-planes determined by the largest eigenvalue of H($e^{i\theta}$A). Adam,
Psarrakos and Tsatsomeros showed that using the two largest eigenvalues of
H(A), the eigenvalues of A satisfy a cubic inequality and the envelope of such
cubic curves defines a region in the complex plane smaller than the numerical
range but still containing the spectrum of A. Here it is shown how using the
three largest eigenvalues of H(A) or more, one obtains new inequalities for the
eigenvalues of A and new envelope-type regions containing the spectrum of A.
