Geodesic
Elementary Proofs of the Extremal Properties of the Eigenvalues of the Sturm-Liouville Equation
Canadian mathematical bulletin, Tome 3 (1960) no. 1, pp. 59-77

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If the Sturm-Liouville eigenvalue problem 1 is first approached from the standpoint of differential equations theory - as opposed, say, to the calculus of variations, or the theory of integral equations - the extremal properties of the eigenvalues seem to be generally regarded as lying beyond the scope of the theory. Thus, neither in the standard work of Bocher [l], nor in the recent work of Coddington and Levinson [2] is any mention made of this topic. Collatz [3, 166-8] gives an elementary proof of the minimum property of the least positive eigenvalue of (1.1), and a brief indication of how this argument can be extended to the higher eigenvalues. The purpose of this paper is to consolidate this elementary approach, and to extend it to cover the singular cases where either the interval is infinite, or one or more of the coefficients are singular at the end-points. 
Beesack, Paul R. Elementary Proofs of the Extremal Properties of the Eigenvalues of the Sturm-Liouville Equation. Canadian mathematical bulletin, Tome 3 (1960) no. 1, pp. 59-77. doi: 10.4153/CMB-1960-010-3
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     title = {Elementary {Proofs} of the {Extremal} {Properties} of the {Eigenvalues} of the {Sturm-Liouville} {Equation}},
     journal = {Canadian mathematical bulletin},
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     year = {1960},
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     doi = {10.4153/CMB-1960-010-3},
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