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• W2009408047 abstract "We prove the following subadditive property of the error function: <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=erf left-parenthesis x right-parenthesis equals StartFraction 2 Over StartRoot pi EndRoot EndFraction integral Subscript 0 Superscript x Baseline e Superscript minus t squared Baseline d t left-parenthesis x element-of bold upper R right-parenthesis period> <mml:semantics> <mml:mrow> <mml:mstyle displaystyle=false scriptlevel=0> <mml:mtext>erf</mml:mtext> </mml:mstyle> <mml:mspace width=thinmathspace /> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>2</mml:mn> <mml:msqrt> <mml:mi>π<!-- π --></mml:mi> </mml:msqrt> </mml:mfrac> <mml:msubsup> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mn>0</mml:mn> <mml:mi>x</mml:mi> </mml:msubsup> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msup> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mspace width=1em /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>R</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mbox {erf},(x)=frac {2}{sqrt {pi }}int _0^x e^{-t^2}dt quad {(xin mathbf {R})}.</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding=application/x-tex>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=b> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding=application/x-tex>b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be real numbers. The inequality <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=erf left-parenthesis left-parenthesis x plus y right-parenthesis Superscript a Baseline right-parenthesis Superscript b Baseline greater-than erf left-parenthesis x Superscript a Baseline right-parenthesis Superscript b Baseline plus erf left-parenthesis y Superscript a Baseline right-parenthesis Superscript b> <mml:semantics> <mml:mrow> <mml:mstyle displaystyle=false scriptlevel=0> <mml:mtext>erf</mml:mtext> </mml:mstyle> <mml:mspace width=thinmathspace /> <mml:mstyle scriptlevel=0> <mml:mrow class=MJX-TeXAtom-OPEN> <mml:mo maxsize=1.2em minsize=1.2em>(</mml:mo> </mml:mrow> </mml:mstyle> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>y</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>a</mml:mi> </mml:msup> <mml:msup> <mml:mstyle scriptlevel=0> <mml:mrow class=MJX-TeXAtom-CLOSE> <mml:mo maxsize=1.2em minsize=1.2em>)</mml:mo> </mml:mrow> </mml:mstyle> <mml:mi>b</mml:mi> </mml:msup> <mml:mo>></mml:mo> <mml:mstyle displaystyle=false scriptlevel=0> <mml:mtext>erf</mml:mtext> </mml:mstyle> <mml:mspace width=thinmathspace /> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>a</mml:mi> </mml:msup> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>b</mml:mi> </mml:msup> <mml:mo>+</mml:mo> <mml:mstyle displaystyle=false scriptlevel=0> <mml:mtext>erf</mml:mtext> </mml:mstyle> <mml:mspace width=thinmathspace /> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mi>a</mml:mi> </mml:msup> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>b</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>mbox {erf},bigl ((x+y)^abigr )^b> mbox {erf},(x^a)^b + mbox {erf},(y^a)^b</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> holds for all positive real numbers <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding=application/x-tex>x</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=y> <mml:semantics> <mml:mi>y</mml:mi> <mml:annotation encoding=application/x-tex>y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a b less-than-or-equal-to 1> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>ableq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W2009408047 title "A subadditive property of the error function" @default.
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