Geodesic
Bounds for Characteristic Values of Positive Definite Matrices
Canadian mathematical bulletin, Tome 15 (1972) no. 1, pp. 51-56

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We consider the problem of determining the best possible bounds on the eigenvalues of an nth order positive definite matrix B, when the determinant (D) and trace (T) are given. A large variety of bounds on the eigenvalues are known when different information concerning B is available (see, for example, [1], [2]). Since D and T simply provide the geometric mean and arithmetic mean of the positive, real eigenvalues of B, the solution to the problem involves certain inequalities satisfied by these means (see [3] for such inequalities in a more general setting). A related problem in which the largest and smallest eigenvalue are known, and inequalities involving D and T are obtained, is described in [4]. 
Binding, P. A.; Hoskins, W. D.; Ponzo, P. J. Bounds for Characteristic Values of Positive Definite Matrices. Canadian mathematical bulletin, Tome 15 (1972) no. 1, pp. 51-56. doi: 10.4153/CMB-1972-011-6
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     author = {Binding, P. A. and Hoskins, W. D. and Ponzo, P. J.},
     title = {Bounds for {Characteristic} {Values} of {Positive} {Definite} {Matrices}},
     journal = {Canadian mathematical bulletin},
     pages = {51--56},
     year = {1972},
     volume = {15},
     number = {1},
     doi = {10.4153/CMB-1972-011-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1972-011-6/}
}
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