Ungleichungen für Mittelwerte. (Inequalities for means). (German) Zbl 0585.26014

Wenn \[ G(x,y)=(xy)^{1/2}\quad das\quad geometrische\quad Mittel, \]
\[ I(x,y)=(1/e)(x^ x/y^ y)^{1/(x-y)}\quad das\quad identric\quad mean, \]
\[ L(x,y)=(x-y)/(\ln x-\ln y)\quad das\quad \log arithmische\quad Mittel, \]
\[ A(x,y)=(x+y)/2\quad das\quad arithmetische\quad Mittel \] der positiven Zahlen x und y (x\(\neq y)\) bezeichnet, dann gilt: \[ (A(x,y)G(x,y))^{1/2}<(L(x,y)I(x,y))^{1/2} \]
\[ <(L(x,y)+I(x,y))<(A(x,y)+G(x,y)) \] für alle positiven Werte x und y mit \(x\neq y\). Wenn mit \(M_ r(x,y)=((x^ r+y^ r)/2)^{1/r}\) für reelle \(r\neq 0\) \(M_ 0(x,y)=(xy)^{1/2}\) das Potenz Mittel von x und y bezeichnet wird, dann gilt für alle positiven Zahlen x und y mit \(x\neq y:\) \[ (*)\quad M_ 0(x,y)<(L(x,y)I(x,y))^{1/2}<M_{1/2}(x,y). \] In (*) kann weder 0 durch einen größeren noch 1/2 durch einen kleineren Wert ersetzt werden.


26D15 Inequalities for sums, series and integrals
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