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

Wenn

G(x,y)=(xy) 1/2 dasgeometrischeMittel,
I(x,y)=(1/e)(x x /y y ) 1/(x-y) dasidentricmean,
L(x,y)=(x-y)/(lnx-lny)daslogarithmischeMittel,
A(x,y)=(x+y)/2dasarithmetischeMittel

der positiven Zahlen x und y (xy) 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 xy. Wenn mit M r (x,y)=((x r +y r )/2) 1/r für reelle r0 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 xy:

(*)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.


MSC:
26D15Inequalities for sums, series and integrals of real functions
References:
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