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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Zaloguj się</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Strony dla anonimowych edytorów <a href="/wiki/Pomoc:Pierwsze_kroki" aria-label="Dowiedz się więcej na temat edytowania"><span>dowiedz się więcej</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Specjalna:M%C3%B3j_wk%C5%82ad" title="Lista edycji wykonanych z tego adresu IP [y]" accesskey="y"><span>Edycje</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Specjalna:Moja_dyskusja" title="Dyskusja użytkownika dla tego adresu IP [n]" accesskey="n"><span>Dyskusja</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Witryna"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Spis treści" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Spis treści</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ukryj</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Początek</div> </a> </li> <li id="toc-Szczególne_przypadki" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Szczególne_przypadki"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Szczególne przypadki</span> </div> </a> <ul id="toc-Szczególne_przypadki-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Spis treści" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Przełącz stan spisu treści" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Przełącz stan spisu treści</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Średnia Stolarskiego</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Przejdź do artykułu w innym języku. Treść dostępna w 6 językach" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-6" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">6 języków</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Stolarsky-Mittel" title="Stolarsky-Mittel – niemiecki" lang="de" hreflang="de" data-title="Stolarsky-Mittel" data-language-autonym="Deutsch" data-language-local-name="niemiecki" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Stolarsky_mean" title="Stolarsky mean – angielski" lang="en" hreflang="en" data-title="Stolarsky mean" data-language-autonym="English" data-language-local-name="angielski" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Moyenne_de_Stolarsky" title="Moyenne de Stolarsky – francuski" lang="fr" hreflang="fr" data-title="Moyenne de Stolarsky" data-language-autonym="Français" data-language-local-name="francuski" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%9E%D7%95%D7%A6%D7%A2_%D7%A1%D7%98%D7%95%D7%9C%D7%A8%D7%A1%D7%A7%D7%99" title="ממוצע סטולרסקי – hebrajski" lang="he" hreflang="he" data-title="ממוצע סטולרסקי" data-language-autonym="עברית" data-language-local-name="hebrajski" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%98%E1%9E%92%E1%9F%92%E1%9E%99%E1%9E%98%E1%9E%9F%E1%9F%92%E1%9E%8F%E1%9E%BC%E1%9E%A1%E1%9E%B6%E1%9E%9A%E1%9E%9F%E1%9F%92%E1%9E%82%E1%9E%B8" title="មធ្យមស្តូឡារស្គី – khmerski" lang="km" hreflang="km" data-title="មធ្យមស្តូឡារស្គី" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmerski" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Stolarsky_ortalamas%C4%B1" title="Stolarsky ortalaması – turecki" lang="tr" hreflang="tr" data-title="Stolarsky ortalaması" data-language-autonym="Türkçe" data-language-local-name="turecki" class="interlanguage-link-target"><span>Türkçe</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q954404#sitelinks-wikipedia" title="Edytuj linki pomiędzy wersjami językowymi" class="wbc-editpage">Edytuj linki</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Przestrzenie nazw"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%C5%9Arednia_Stolarskiego" title="Zobacz stronę treści [c]" accesskey="c"><span>Artykuł</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Dyskusja:%C5%9Arednia_Stolarskiego&action=edit&redlink=1" rel="discussion" class="new" title="Dyskusja o zawartości tej strony (strona nie istnieje) [t]" accesskey="t"><span>Dyskusja</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Zmień wariant języka" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">polski</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Widok"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%C5%9Arednia_Stolarskiego"><span>Czytaj</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%C5%9Arednia_Stolarskiego&veaction=edit" title="Edytuj tę stronę [v]" accesskey="v"><span>Edytuj</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%C5%9Arednia_Stolarskiego&action=edit" title="Edycja kodu źródłowego strony [e]" accesskey="e"><span>Edytuj kod źródłowy</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%C5%9Arednia_Stolarskiego&action=history" title="Starsze wersje tej strony [h]" accesskey="h"><span>Wyświetl historię</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Narzędzia dla stron"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Narzędzia" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Narzędzia</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Narzędzia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ukryj</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Więcej opcji" > <div class="vector-menu-heading"> Działania </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%C5%9Arednia_Stolarskiego"><span>Czytaj</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%C5%9Arednia_Stolarskiego&veaction=edit" title="Edytuj tę stronę [v]" accesskey="v"><span>Edytuj</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%C5%9Arednia_Stolarskiego&action=edit" title="Edycja kodu źródłowego strony [e]" accesskey="e"><span>Edytuj kod źródłowy</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%C5%9Arednia_Stolarskiego&action=history"><span>Wyświetl historię</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Ogólne </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Specjalna:Linkuj%C4%85ce/%C5%9Arednia_Stolarskiego" title="Pokaż listę wszystkich stron linkujących do tej strony [j]" accesskey="j"><span>Linkujące</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Specjalna:Zmiany_w_linkowanych/%C5%9Arednia_Stolarskiego" rel="nofollow" title="Ostatnie zmiany w stronach, do których ta strona linkuje [k]" accesskey="k"><span>Zmiany w linkowanych</span></a></li><li id="t-upload" class="mw-list-item"><a href="//pl.wikipedia.org/wiki/Wikipedia:Prześlij_plik" title="Prześlij pliki [u]" accesskey="u"><span>Prześlij plik</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Specjalna:Strony_specjalne" title="Lista wszystkich stron specjalnych [q]" accesskey="q"><span>Strony specjalne</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%C5%9Arednia_Stolarskiego&oldid=70555429" title="Stały link do tej wersji tej strony"><span>Link do tej wersji</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%C5%9Arednia_Stolarskiego&action=info" title="Więcej informacji 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title="Średnia">średnia</a>, której szczególnymi przypadkami jest wiele klasycznych średnich, zdefiniowana dla ustalonego parametru <i>p</i> oraz dodatnich argumentów wzorem </p> <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 S_{p}(x,y)={\begin{cases}x&{\text{jeśli }}x=y\\\left({\frac {x^{p}-y^{p}}{p(x-y)}}\right)^{\frac {1}{p-1}}&{\text{wpp.}}\end{cases}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>jeśli </mtext> </mrow> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mrow> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </msup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>wpp.</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{p}(x,y)={\begin{cases}x&{\text{jeśli }}x=y\\\left({\frac {x^{p}-y^{p}}{p(x-y)}}\right)^{\frac {1}{p-1}}&{\text{wpp.}}\end{cases}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c85a0d9a6ac2b90df53dcfdae78b305c1742649" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.806ex; margin-bottom: -0.199ex; width:39.515ex; height:9.176ex;" alt="{\displaystyle S_{p}(x,y)={\begin{cases}x&{\text{jeśli }}x=y\\\left({\frac {x^{p}-y^{p}}{p(x-y)}}\right)^{\frac {1}{p-1}}&{\text{wpp.}}\end{cases}}.}"></span></dd></dl> <p>Gdzie parametr <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 p\in \mathbb {R} /\{0,1\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {R} /\{0,1\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec4559547db5d0d920b857e20731deea9f518c52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:13.271ex; height:2.843ex;" alt="{\displaystyle p\in \mathbb {R} /\{0,1\}.}"></span> </p><p>Można pokazać, że tak zdefiniowana funkcja jest średnią jej argumentów stosując <a href="/wiki/Twierdzenie_Lagrange%E2%80%99a_(rachunek_r%C3%B3%C5%BCniczkowy)" title="Twierdzenie Lagrange’a (rachunek różniczkowy)">twierdzenia Lagrange’a</a> dla liczb x i y oraz funkcji <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(x)=x^{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x^{p}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f53c4baae9c1da03ae9bc658869f8859d841ee9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.552ex; height:2.843ex;" alt="{\displaystyle f(x)=x^{p}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Szczególne_przypadki"><span id="Szczeg.C3.B3lne_przypadki"></span>Szczególne przypadki</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C5%9Arednia_Stolarskiego&veaction=edit&section=1" title="Edytuj sekcję: Szczególne przypadki" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C5%9Arednia_Stolarskiego&action=edit&section=1" title="Edytuj kod źródłowy sekcji: Szczególne przypadki"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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 \lim _{p\to -\infty }S_{p}(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">→</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{p\to -\infty }S_{p}(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a50f468ab629b691193d9b0da06e6e06b2b1c38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; margin-left: -0.063ex; width:13.655ex; height:4.176ex;" alt="{\displaystyle \lim _{p\to -\infty }S_{p}(x,y)}"></span> jest <a href="/wiki/Funkcje_minimum_i_maksimum" title="Funkcje minimum i maksimum">minimum</a>.</li> <li><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 S_{-1}(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{-1}(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ba3ec6cc8a510fddcc4a1ce211133c93b1ea4ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.086ex; height:2.843ex;" alt="{\displaystyle S_{-1}(x,y)}"></span> jest <a href="/wiki/%C5%9Arednia_geometryczna" title="Średnia geometryczna">średnią geometryczną</a>.</li> <li><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 \lim _{p\to 0}S_{p}(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{p\to 0}S_{p}(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66d2390ad0e599b5ac61ef00e6e3b153aafc24f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.063ex; width:11.555ex; height:4.343ex;" alt="{\displaystyle \lim _{p\to 0}S_{p}(x,y)}"></span> jest <a href="/wiki/%C5%9Arednia_logarytmiczna" title="Średnia logarytmiczna">średnią logarytmiczną</a>.</li> <li><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 S_{\frac {1}{2}}(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\frac {1}{2}}(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad45bbaf7a37639384775434b63b83c80d4e094" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:8.489ex; height:4.176ex;" alt="{\displaystyle S_{\frac {1}{2}}(x,y)}"></span> jest <a href="/wiki/%C5%9Arednia_pot%C4%99gowa" title="Średnia potęgowa">średnią potęgową</a> dla wykładnika ½.</li> <li><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 S_{2}(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{2}(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f83abecbd1c49c7b8ebad346b4645c43af7ebc7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.808ex; height:2.843ex;" alt="{\displaystyle S_{2}(x,y)}"></span> jest <a href="/wiki/%C5%9Arednia_arytmetyczna" title="Średnia arytmetyczna">średnią arytmetyczną</a>.</li> <li><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 \lim _{p\to \infty }S_{p}(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{p\to \infty }S_{p}(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bf5fa7a728668e235098afa82975e6b0b4813d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; margin-left: -0.063ex; width:12.376ex; height:4.176ex;" alt="{\displaystyle \lim _{p\to \infty }S_{p}(x,y)}"></span> jest <a href="/wiki/Funkcje_minimum_i_maksimum" title="Funkcje minimum i maksimum">maksimum</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C5%9Arednia_Stolarskiego&veaction=edit&section=2" title="Edytuj sekcję: Bibliografia" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C5%9Arednia_Stolarskiego&action=edit&section=2" title="Edytuj kod źródłowy sekcji: Bibliografia"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Stolarsky, Kenneth B.: <i><a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2689825">Generalizations of the logarithmic mean</a></i>, Mathematics Magazine, Vol. 48, No. 2, Mar., 1975, pp 87-92</li> <li><cite class="citation open-access"><span class="cite-name-before"><span class="cite-name-full">Eric W.</span><span class="cite-name-initials" title="Eric W." style="display:none">E.W.</span> </span><span class="cite-lastname">Weisstein</span><span class="cite-name-after" style="display:none"> <span class="cite-name-full">Eric W.</span><span class="cite-name-initials" title="Eric W.">E.W.</span></span>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/StolarskyMean.html"><i>Stolarsky Mean</i></a>, [w:] <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>, <a href="/wiki/Wolfram_Research" title="Wolfram Research">Wolfram Research</a><span class="accessdate"> [dostęp 2020-12-13]</span><span class="Z3988" 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