Geodesic
Estimates for the Eigenvalues of Matrices
Matematičeskie zametki, Tome 79 (2006) no. 2, pp. 169-177.

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For matrices whose eigenvalues are real (such as Hermitian or real symmetric matrices), we derive simple explicit estimates for the maximal $(\lambda_{\max})$ and the minimal $(\lambda_{\min})$ eigenvalues in terms of determinants of order less than 3. For $3\times3$ matrices, we derive sharper estimates, which use $\det A$ but do not require to solve cubic equations. 
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R. I. Alidema; A. F. Filippov. Estimates for the Eigenvalues of Matrices. Matematičeskie zametki, Tome 79 (2006) no. 2, pp. 169-177. http://geodesic.mathdoc.fr/item/MZM_2006_79_2_a1/

[1] Fadeev D. K., Fadeeva V. N., Vychislitelnye metody lineinoi algebry, Fizmatgiz, M., 1960

[2] Gerschgorin S., “Über die Abgrenzung der Eigenwerte einer Matrix”, Izv. AN SSSR. Matem. i estestv. nauki, 1931, no. 7, 749–754

[3] Bakhvalov N. S., Zhadkov N. P., Kobelkov G. M., Chislennye metody, Nauka, M., 1987 | Zbl