@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr: a skos:ConceptScheme .
psr:-SKTRS1V0-R
  skos:prefLabel "real analysis"@en, "analyse réelle"@fr ;
  a skos:Concept ;
  skos:narrower psr:-W8ZW5NCL-4 .

psr:-W42D202L-K
  skos:prefLabel "inégalité"@fr, "inequality"@en ;
  a skos:Concept ;
  skos:narrower psr:-W8ZW5NCL-4 .

psr:-W8ZW5NCL-4
  dc:created "2023-08-11"^^xsd:date ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/In%C3%A9galit%C3%A9_de_Steffensen>, <https://en.wikipedia.org/wiki/Steffensen%27s_inequality> ;
  skos:prefLabel "inégalité de Steffensen"@fr, "Steffensen's inequality"@en ;
  skos:broader psr:-W42D202L-K, psr:-SKTRS1V0-R ;
  skos:definition """En mathématiques, et plus précisément en analyse réelle, l'<b>inégalité de Steffensen</b> est une inégalité nommée d'après Johan Frederik Steffensen. 
<br/>L'inégalité intégrale est la suivante 
<br/>
<br/><dl><dd>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f:[a,b]\\\\longrightarrow \\\\mathbb {R} }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>        <mo>:</mo>
<br/>        <mo stretchy="false">[</mo>
<br/>        <mi>a</mi>
<br/>        <mo>,</mo>
<br/>        <mi>b</mi>
<br/>        <mo stretchy="false">]</mo>
<br/>        <mo stretchy="false">⟶<!-- ⟶ --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">R</mi>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f:[a,b]\\\\longrightarrow \\\\mathbb {R} }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb09447ab86aaa385a488f810ef8bba56b4b467" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:14.545ex; height:2.843ex;" alt="{\\\\displaystyle f:[a,b]\\\\longrightarrow \\\\mathbb {R} }"></span> est une fonction intégrable positive, décroissante</dd>
<br/><dd>et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle g:[a,b]\\\\longrightarrow [0,1]}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>g</mi>
<br/>        <mo>:</mo>
<br/>        <mo stretchy="false">[</mo>
<br/>        <mi>a</mi>
<br/>        <mo>,</mo>
<br/>        <mi>b</mi>
<br/>        <mo stretchy="false">]</mo>
<br/>        <mo stretchy="false">⟶<!-- ⟶ --></mo>
<br/>        <mo stretchy="false">[</mo>
<br/>        <mn>0</mn>
<br/>        <mo>,</mo>
<br/>        <mn>1</mn>
<br/>        <mo stretchy="false">]</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle g:[a,b]\\\\longrightarrow [0,1]}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a4ecec97ed883366bf9093da2fb7f641f86fa43" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:17.357ex; height:2.843ex;" alt="{\\\\displaystyle g:[a,b]\\\\longrightarrow [0,1]}"></span>est une autre fonction intégrable, alors
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\int _{b-k}^{b}f(x)\\\\,dx\\\\leq \\\\int _{a}^{b}f(x)g(x)\\\\,dx\\\\leq \\\\int _{a}^{a+k}f(x)\\\\,dx,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msubsup>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>b</mi>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi>k</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>b</mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mi>d</mi>
<br/>        <mi>x</mi>
<br/>        <mo>≤<!-- ≤ --></mo>
<br/>        <msubsup>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>a</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>b</mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mi>g</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mi>d</mi>
<br/>        <mi>x</mi>
<br/>        <mo>≤<!-- ≤ --></mo>
<br/>        <msubsup>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>a</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>a</mi>
<br/>            <mo>+</mo>
<br/>            <mi>k</mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mi>d</mi>
<br/>        <mi>x</mi>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\int _{b-k}^{b}f(x)\\\\,dx\\\\leq \\\\int _{a}^{b}f(x)g(x)\\\\,dx\\\\leq \\\\int _{a}^{a+k}f(x)\\\\,dx,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e805530d43ddeec88195fcc29537d723ec037ec" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:47.779ex; height:6.509ex;" alt="{\\\\displaystyle \\\\int _{b-k}^{b}f(x)\\\\,dx\\\\leq \\\\int _{a}^{b}f(x)g(x)\\\\,dx\\\\leq \\\\int _{a}^{a+k}f(x)\\\\,dx,}"></span></dd></dl></dd>
<br/><dd>où
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle k=\\\\int _{a}^{b}g(x)\\\\,dx.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>k</mi>
<br/>        <mo>=</mo>
<br/>        <msubsup>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>a</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>b</mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mi>g</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mi>d</mi>
<br/>        <mi>x</mi>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle k=\\\\int _{a}^{b}g(x)\\\\,dx.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da0edc728262b4a6ff856926b42d0aa643d7825b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:15.933ex; height:6.343ex;" alt="{\\\\displaystyle k=\\\\int _{a}^{b}g(x)\\\\,dx.}"> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/In%C3%A9galit%C3%A9_de_Steffensen">https://fr.wikipedia.org/wiki/In%C3%A9galit%C3%A9_de_Steffensen</a>)"""@fr, """<b>Steffensen's inequality</b> is an equation in mathematics named after Johan Frederik Steffensen.  
<br/>It is an integral inequality in real analysis, stating: 
<br/>
<br/><dl><dd>If ƒ&nbsp;:&nbsp;[<i>a</i>,&nbsp;<i>b</i>]&nbsp;→&nbsp;<b>R</b> is a non-negative, monotonically decreasing, integrable function</dd>
<br/><dd>and <i>g</i>&nbsp;:&nbsp;[<i>a</i>,&nbsp;<i>b</i>]&nbsp;→&nbsp;[0,&nbsp;1] is another integrable function, then</dd></dl>
<br/><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\int _{b-k}^{b}f(x)\\\\,dx\\\\leq \\\\int _{a}^{b}f(x)g(x)\\\\,dx\\\\leq \\\\int _{a}^{a+k}f(x)\\\\,dx,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msubsup>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>b</mi>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi>k</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>b</mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mi>d</mi>
<br/>        <mi>x</mi>
<br/>        <mo>≤<!-- ≤ --></mo>
<br/>        <msubsup>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>a</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>b</mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mi>g</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mi>d</mi>
<br/>        <mi>x</mi>
<br/>        <mo>≤<!-- ≤ --></mo>
<br/>        <msubsup>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>a</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>a</mi>
<br/>            <mo>+</mo>
<br/>            <mi>k</mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mi>f</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mi>d</mi>
<br/>        <mi>x</mi>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\int _{b-k}^{b}f(x)\\\\,dx\\\\leq \\\\int _{a}^{b}f(x)g(x)\\\\,dx\\\\leq \\\\int _{a}^{a+k}f(x)\\\\,dx,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e805530d43ddeec88195fcc29537d723ec037ec" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:47.779ex; height:6.509ex;" alt="{\\\\displaystyle \\\\int _{b-k}^{b}f(x)\\\\,dx\\\\leq \\\\int _{a}^{b}f(x)g(x)\\\\,dx\\\\leq \\\\int _{a}^{a+k}f(x)\\\\,dx,}"></span></dd></dl></dd>
<br/><dd>where
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle k=\\\\int _{a}^{b}g(x)\\\\,dx.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>k</mi>
<br/>        <mo>=</mo>
<br/>        <msubsup>
<br/>          <mo>∫<!-- ∫ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>a</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>b</mi>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mi>g</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mi>d</mi>
<br/>        <mi>x</mi>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle k=\\\\int _{a}^{b}g(x)\\\\,dx.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da0edc728262b4a6ff856926b42d0aa643d7825b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:15.933ex; height:6.343ex;" alt="{\\\\displaystyle k=\\\\int _{a}^{b}g(x)\\\\,dx.}"> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Steffensen%27s_inequality">https://en.wikipedia.org/wiki/Steffensen%27s_inequality</a>)"""@en ;
  dc:modified "2023-08-11"^^xsd:date ;
  skos:inScheme psr: ;
  a skos:Concept .