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Inequalities for elementary means. (Ungleichungen für elementare Mittelwerte.) (German) Zbl 0813.26009
Für die verschiedenen positiven Zahlen \(x\) und \(y\) seien \(G(x, y):= \sqrt{xy}\), \(A(x, y):= (x+ y)/2\), \(L(x, y):= (x- y)/(\ln x- \ln y)\) und \(I(x, y):= e^{-1}(x^ x/ y^ y)^{1/(x- y)}\). Gezeigt werden die Ungleichungen \[ L(x, y)< \sqrt{L(G(x, y)^ 2, A(x,y)^ 2)}< \sqrt{I(G(x, y)^ 2, A(x, y)^ 2)}< I(x, y). \] Die rechte Seite dieser Ungleichungskette sowie einige weitere Ungleichungen, werden unter Benutzung von Identitäten bewiesen.

26D15 Inequalities for sums, series and integrals
inequalities; means
Full Text: DOI
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