Sqrt[a b] <= ArithmeticGeometricMean[a, b] <= (a + b)/2

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <msqrt> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </msqrt> <mo> &#8804; </mo> <mrow> <semantics> <mi> agm </mi> <annotation-xml encoding='MathML-Content'> <ci> ArithmeticGeometricMean </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <leq /> <apply> <power /> <apply> <times /> <ci> a </ci> <ci> b </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> ArithmeticGeometricMean </ci> <ci> a </ci> <ci> b </ci> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[SqrtBox[RowBox[List["a", " ", "b"]]], "\[LessEqual]", RowBox[List["ArithmeticGeometricMean", "[", RowBox[List["a", ",", "b"]], "]"]], "\[LessEqual]", FractionBox[RowBox[List["a", "+", "b"]], "2"]]]]]

2001-10-29