Geodesic
A proof of monotony of the Temple quotients in eigenvalue problems
Applications of Mathematics, Tome 29 (1984) no. 2, pp. 149-158
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If the so-called Collatz method is applied to get twosided estimates of the first eigenvalue $\lambda_1$, the sequences of the so-called Schwarz quatients (which are upper bounds for $\lambda_1$) and of the so-called Temple quotients (which are lower bounds) are constructed. While monotony of the first sequence was proved many years ago, monotony of the second one has been proved only recently by F. goerisch and J. Albrecht in their common paper "Die Monotonie der Templeschen Quotienten" (ZAMM, in print). In the present paper another (so to say elementary) proof is given.
If the so-called Collatz method is applied to get twosided estimates of the first eigenvalue $\lambda_1$, the sequences of the so-called Schwarz quatients (which are upper bounds for $\lambda_1$) and of the so-called Temple quotients (which are lower bounds) are constructed. While monotony of the first sequence was proved many years ago, monotony of the second one has been proved only recently by F. goerisch and J. Albrecht in their common paper "Die Monotonie der Templeschen Quotienten" (ZAMM, in print). In the present paper another (so to say elementary) proof is given.
DOI : 10.21136/AM.1984.104078
Classification : 34L99, 35P15, 65L15, 65N25
Keywords: monotony; Collatz method; first eigenvalue; Schwarz quotients; Temple quotients 
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Rektorys, Karel. A proof of monotony of the Temple quotients in eigenvalue problems. Applications of Mathematics, Tome 29 (1984) no. 2, pp. 149-158. doi: 10.21136/AM.1984.104078

[1] F. Goerisch J. Albrecht: Die Mononie der Templeschen Quotienten. ZAMM (in print).

[2] K. Rektorys: Variational Methods in Mathematics, Science and Engineering. 2nd Ed. Dordrecht- Boston-London, J. Reidel 1979. (Czech: Praha, SNTL 1974.) | MR

[3] K. Rektorys Z. Vospěl: On a method of twosided eigenvalue estimates for elliptic equations of the form $Au - \lambda Bu = 0$. Aplikace matematiky 26 (1981), 211-240. | MR

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