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Ungleichungen für Mittelwerte. (Inequalities for means). (German) Zbl 0585.26014
Wenn $$ G(x,y)=(xy)\sp{1/2}\quad das\quad geometrische\quad Mittel, $$ $$ I(x,y)=(1/e)(x\sp x/y\sp y)\sp{1/(x-y)}\quad das\quad identric\quad mean, $$ $$ L(x,y)=(x-y)/(\ln x-\ln y)\quad das\quad \log arithmische\quad Mittel, $$ $$ A(x,y)=(x+y)/2\quad das\quad arithmetische\quad Mittel $$ der positiven Zahlen x und y (x$\ne y)$ bezeichnet, dann gilt: $$ (A(x,y)G(x,y))\sp{1/2}<(L(x,y)I(x,y))\sp{1/2} $$ $$ <(L(x,y)+I(x,y))<(A(x,y)+G(x,y)) $$ für alle positiven Werte x und y mit $x\ne y$. Wenn mit $M\sb r(x,y)=((x\sp r+y\sp r)/2)\sp{1/r}$ für reelle $r\ne 0$ $M\sb 0(x,y)=(xy)\sp{1/2}$ das Potenz Mittel von x und y bezeichnet wird, dann gilt für alle positiven Zahlen x und y mit $x\ne y:$ $$ (*)\quad M\sb 0(x,y)<(L(x,y)I(x,y))\sp{1/2}<M\sb{1/2}(x,y). $$ In (*) kann weder 0 durch einen größeren noch 1/2 durch einen kleineren Wert ersetzt werden.

26D15Inequalities for sums, series and integrals of real functions
Full Text: DOI
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