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Some remarks on a certain question about the minimum. (Quelques remarques sur une certaine question de minimum.) (French) JFM 18.0325.01
Eine von Markoff in seinem Werke “Sur quelques applications des fractions continues algébriques”. St. Petersburg 1884 (s. F. d. M. XVII. 168, JFM 17.0168.01) gelöste Aufgabe wird durch teilweise Integration und einige Substitutionen in folgende überfḧrt, deren Lösung sich mithin auf jene reducirt. Von der unbekannten Function $f(x)$, welche, wenn $x$ von 0 bis 1 wächst, beständig abnimmt, sind die Werte $$f(1), \quad \int_0^1 x^2 f(x) dx, \quad \int_0^1 x^4 f(x)dx, \quad \dots \int_0^1 x^{2n} f(x) dx = a_{2n}$$ gegeben, man soll die genaue untere Grenze von $f(0)$ finden. Es wird schliesslich erhalten: $$f(0) \geqq f(1) + \sqrt{\frac{[3a_2 - f(1)]^5}{[3a_4 - f(1)]^3}}$$ übereinstimmend mit einem Resultat von Stieltjes.
Reviewer: Hoppe, Prof. (Berlin)
44A60Moment problems (integral transforms)
49K05Free problems in one independent variable (optimality conditions)
Full Text: DOI EuDML