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W2171804887 abstract "For <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A subset-of upper X equals StartSet x 1 comma period period period comma x Subscript n Baseline vertical-bar x Subscript i Baseline greater-than-or-equal-to 0 comma i equals 1 comma 2 comma period period period comma n EndSet comma> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>.</mml:mo> <mml:mo>.</mml:mo> <mml:mo>.</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mtext> </mml:mtext> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mtext> </mml:mtext> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>.</mml:mo> <mml:mo>.</mml:mo> <mml:mo>.</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Asubset X={x_1,...,x_n ~|~x_igeq 0, ~i=1,2,...,n},</mml:annotation> </mml:semantics> </mml:math> </inline-formula> let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a Subscript upper A> <mml:semantics> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>A</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>a_A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g Subscript upper A> <mml:semantics> <mml:msub> <mml:mi>g</mml:mi> <mml:mi>A</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>g_A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the arithmetic mean and geometric mean of elements of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, respectively. It is proved that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an integer in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis StartFraction n Over 2 EndFraction comma n right-bracket> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mfrac> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(frac {n}{2}, n]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis product Underscript StartLayout 1st Row StartAbsoluteValue upper A EndAbsoluteValue equals k 2nd Row upper A subset-of upper X EndLayout Endscripts a Subscript upper A Baseline right-parenthesis Superscript StartFraction 1 Over upper C Super Subscript n Super Superscript k Superscript EndFraction Baseline greater-than-or-equal-to StartFraction 1 Over upper C Subscript n Superscript k Baseline EndFraction left-parenthesis sigma-summation Underscript StartLayout 1st Row StartAbsoluteValue upper A EndAbsoluteValue equals k 2nd Row upper A subset-of upper X EndLayout Endscripts g Subscript upper A Baseline right-parenthesis comma> <mml:semantics> <mml:mrow> <mml:mstyle scriptlevel=0> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo maxsize=1.623em minsize=1.623em>(</mml:mo> </mml:mrow> </mml:mstyle> <mml:munder> <mml:mo>∏</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mfrac linethickness=0> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>A</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:munder> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> <mml:msup> <mml:mstyle scriptlevel=0> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo maxsize=1.623em minsize=1.623em>)</mml:mo> </mml:mrow> </mml:mstyle> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>n</mml:mi> <mml:mi>k</mml:mi> </mml:msubsup> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo>≥</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>n</mml:mi> <mml:mi>k</mml:mi> </mml:msubsup> </mml:mfrac> <mml:mstyle scriptlevel=0> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo maxsize=1.623em minsize=1.623em>(</mml:mo> </mml:mrow> </mml:mstyle> <mml:munder> <mml:mo>∑</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mfrac linethickness=0> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>A</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:munder> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> <mml:mstyle scriptlevel=0> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo maxsize=1.623em minsize=1.623em>)</mml:mo> </mml:mrow> </mml:mstyle> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Big (prod _{|A|=katop Asubset X}a_{A}Big )^{frac {1}{C_n^k}} geq frac {1}{C_n^k}Big (sum _{|A|=katop Asubset X}g_{A}Big ),</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> with equality if and only if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x 1 equals period period period equals x Subscript n Baseline> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>.</mml:mo> <mml:mo>.</mml:mo> <mml:mo>.</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>x_1=...=x_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Furthermore, as a generalization of this inequality, a mixed power-mean inequality for subsets is established." @default.
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W2171804887 date "2004-03-24" @default.
W2171804887 modified "2023-09-24" @default.
W2171804887 title "Mixed-mean inequalities for subsets" @default.
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