GermanWikipedia - /page/entry.mu

<!DOCTYPE html>

<title>Additive Funktion</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Additive Funktion</span></h1>

</header>

</div>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p><b>Additive</b>, <b>subadditive</b> und <b>superadditive Funktionen</b> sind mathematische Objekte. Es sind bestimmte Klassen von <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktionen</a>. <a href="Lineare_Abbildung" title="Lineare Abbildung">Lineare Abbildungen</a> sind besondere additive Funktionen.

</p><p>In der <a href="Zahlentheorie" title="Zahlentheorie">Zahlentheorie</a> herrscht eine andere Definition für die <i>additive Funktion</i>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Eine Funktion <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> heißt additiv, wenn sie die <a href="Funktionalgleichung" title="Funktionalgleichung">Funktionalgleichung</a>

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x+y)=f(x)+f(y)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11e072f8427aa606b95bad4d8fba9cb3da2c0b09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.013ex; height:2.843ex;" alt="{\displaystyle f(x+y)=f(x)+f(y)}" loading="lazy"></span></dd></dl>

<p>erfüllt.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Sind <a href="Definitionsmenge" title="Definitionsmenge">Definitions-</a> und <a href="Zielmenge" title="Zielmenge">Zielbereich</a> <a href="Abelsche_Gruppe" title="Abelsche Gruppe">abelsche Gruppen</a>, so spricht man auch von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"></span>-<a href="Lineare_Abbildung" title="Lineare Abbildung">Linearität</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Sub-_und_Superadditive_Funktionen">Sub- und Superadditive Funktionen</h2></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> eine <a href="Halbgruppe" title="Halbgruppe">Halbgruppe</a> mit der <a href="Zweistellige_Verkn%C3%BCpfung" title="Zweistellige Verknüpfung">Verknüpfung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" loading="lazy"></span>, so heißt eine Abbildung <span class="mwe-math-element mwe-math-element-inline"><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\colon M\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon M\to \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b28c87bb68bdeda0931c6fa7210a19a7da47528a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.047ex; height:2.509ex;" alt="{\displaystyle f\colon M\to \mathbb {R} }" loading="lazy"></span> <i>subadditiv</i>, wenn für alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> aus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> gilt:<sup id="cite_ref-ConvexFunctions_2-0" class="reference"><a href="#cite_note-ConvexFunctions-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x+y)\leq f(x)+f(y)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0440ac144d656f8aa4edb50a971376aad15ff3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.013ex; height:2.843ex;" alt="{\displaystyle f(x+y)\leq f(x)+f(y)}" loading="lazy"></span>.</dd></dl>

<p>Die Abbildung heißt <i>superadditiv</i>, wenn für alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span> aus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> gilt:<sup id="cite_ref-ConvexFunctions_2-1" class="reference"><a href="#cite_note-ConvexFunctions-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x+y)\geq f(x)+f(y)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8954e8ec8c01d9ec52e82ed8f3cf8d7c31d6f186.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.013ex; height:2.843ex;" alt="{\displaystyle f(x+y)\geq f(x)+f(y)}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>

<ul><li>Gemäß der <a href="Dreiecksungleichung" title="Dreiecksungleichung">Dreiecksungleichung</a> sind <a href="Norm_(Mathematik)" title="Norm (Mathematik)">Normen</a> und <a href="Absoluter_Betrag" class="mw-redirect" title="Absoluter Betrag">Beträge</a> stets subadditiv.</li>

<li><a href="Sublineare_Funktion" title="Sublineare Funktion">Sublineare Funktionen</a> sind subadditiv.</li>

<li><a href="Lineare_Abbildung" title="Lineare Abbildung">Lineare Abbildungen</a> sind additiv.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>

<ul><li>Eine Abbildung ist genau dann additiv, wenn sie sowohl sub- als auch superadditiv ist.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> eine additive Funktion, so gilt für jede endliche Anzahl <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},\dotsc ,x_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{1},\dotsc ,x_{n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/043a9851e76aa50d9939db2bcdd6718c3c325743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle x_{1},\dotsc ,x_{n}}" loading="lazy"></span> von Elementen aus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span>:</li></ul>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x_{1}+\dotsb +x_{n})=f(x_{1})+\dotsb +f(x_{n})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34e21749b7c6e71fb55362bd0d7243b97e3fcacf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.034ex; height:2.843ex;" alt="{\displaystyle f(x_{1}+\dotsb +x_{n})=f(x_{1})+\dotsb +f(x_{n})}" loading="lazy"></span></dd></dl></dd>

<dd>Entsprechendes gilt für Sub- und Superadditivität.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Definition_in_der_Zahlentheorie">Definition in der Zahlentheorie</h2></div>

<p>Bei <a href="Zahlentheoretische_Funktion" title="Zahlentheoretische Funktion">zahlentheoretischen Funktionen</a> <span class="mwe-math-element mwe-math-element-inline"><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\colon \mathbb {N} \to \mathbb {C} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {N} \to \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/595c8b528a92771afe091895547d1780439d5d16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.283ex; height:2.509ex;" alt="{\displaystyle f\colon \mathbb {N} \to \mathbb {C} }" loading="lazy"></span> betrachtet man als Verknüpfung auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdf9a96b565ea202d0f4322e9195613fb26a9bed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" loading="lazy"></span> die Multiplikation. Eine zahlentheoretische Funktion heißt <i>additiv</i>, wenn die Gleichung

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(xy)=f(x)+f(y)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7545d75e5c137551997e84a079dec421b9952f79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.173ex; height:2.843ex;" alt="{\displaystyle f(xy)=f(x)+f(y)}" loading="lazy"></span></dd></dl>

<p>für alle <a href="Teilerfremd" class="mw-redirect" title="Teilerfremd">teilerfremden</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\in \mathbb {N} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc8faeb9dc5e6335420ae9010ecff8c928944ffe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.674ex; height:2.509ex;" alt="{\displaystyle y\in \mathbb {N} }" loading="lazy"></span> gilt. Gilt dies sogar für alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span>, so heißt die Funktion <i>streng additiv</i>.

</p><p>Eine ähnliche Einschränkung der Additivität (auf disjunkte statt beliebige Vereinigungen) gibt es in der Maßtheorie.

</p>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="%CE%A3-Subadditivit%C3%A4t" title="Σ-Subadditivität">σ-Subadditivität</a></li>

<li><a href="%CE%A3-Additivit%C3%A4t" title="Σ-Additivität">σ-Additivität</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Prasanna Sahoo, Thomas Riedel: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Mean Value Theorems and Functional Equations</cite>. 1998, ISBN 981-02-3544-5, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>1</span> (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Additive+Funktion&rft.au=Prasanna+Sahoo%2C+Thomas+Riedel&rft.btitle=Mean+Value+Theorems+and+Functional+Equations&rft.date=1998&rft.genre=book&rft.isbn=9810235445&rft.pages=1" style="display:none"> </span></span>

</li>

<li id="cite_note-ConvexFunctions-2"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-ConvexFunctions_2-0">a</a></sup> <sup><a href="#cite_ref-ConvexFunctions_2-1">b</a></sup></span> <span class="reference-text">Josip E. Peajcariaac, Y. L. Tong: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Convex Functions, Partial Orderings, and Statistical Applications</cite>. Academic Press, 1992, ISBN 0-12-549250-2, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>8</span> (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Additive+Funktion&rft.au=Josip+E.+Peajcariaac%2C+Y.+L.+Tong&rft.btitle=Convex+Functions%2C+Partial+Orderings%2C+and+Statistical+Applications&rft.date=1992&rft.genre=book&rft.isbn=0125492502&rft.pages=8&rft.pub=Academic+Press" style="display:none"> </span></span>

</li>

</ol></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2024-10-01" href="https://de.wikipedia.org/wiki/?title=Additive_Funktion&oldid=249051490">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

</div>

</div></div>

</div>

</main>

</div>

</body></html>