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</div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <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="Toggle the table of contents" > <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">Toggle the table of contents</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">Otto Hölder</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="Go to an article in another language. Available in 24 languages" > <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-24" 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">24 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Otto_H%C3%B6lder" title="Otto Hölder – Azerbaijani" lang="az" hreflang="az" data-title="Otto Hölder" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Otto_H%C3%B6lder" title="Otto Hölder – Catalan" lang="ca" hreflang="ca" data-title="Otto Hölder" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Otto_H%C3%B6lder" title="Otto Hölder – German" lang="de" hreflang="de" data-title="Otto Hölder" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%8C%CF%84%CF%84%CE%BF_%CE%A7%CE%AD%CE%BB%CE%BD%CF%84%CE%B5%CF%81" title="Όττο Χέλντερ – Greek" lang="el" hreflang="el" data-title="Όττο Χέλντερ" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Otto_H%C3%B6lder" title="Otto Hölder – Spanish" lang="es" hreflang="es" data-title="Otto Hölder" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Otto_H%C3%B6lder" title="Otto Hölder – French" lang="fr" hreflang="fr" data-title="Otto Hölder" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%98%A4%ED%86%A0_%ED%9A%94%EB%8D%94" title="오토 횔더 – Korean" lang="ko" hreflang="ko" data-title="오토 횔더" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%91%E0%A4%9F%E0%A5%8B_%E0%A4%B9%E0%A5%8B%E0%A4%B2%E0%A5%8D%E0%A4%A1%E0%A4%B0" title="ऑटो होल्डर – Hindi" lang="hi" hreflang="hi" data-title="ऑटो होल्डर" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Otto_H%C3%B6lder" title="Otto Hölder – Indonesian" lang="id" hreflang="id" data-title="Otto Hölder" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Otto_H%C3%B6lder" title="Otto Hölder – Italian" lang="it" hreflang="it" data-title="Otto Hölder" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%A7%D9%88%D8%AA%D9%88_%D9%87%D9%88%D9%84%D8%AF%D8%B1" title="اوتو هولدر – Egyptian Arabic" lang="arz" hreflang="arz" data-title="اوتو هولدر" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Otto_H%C3%B6lder" title="Otto Hölder – Dutch" lang="nl" hreflang="nl" data-title="Otto Hölder" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AA%E3%83%83%E3%83%88%E3%83%BC%E3%83%BB%E3%83%98%E3%83%AB%E3%83%80%E3%83%BC" title="オットー・ヘルダー – Japanese" lang="ja" hreflang="ja" data-title="オットー・ヘルダー" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Otto_Ludwig_H%C3%B6lder" title="Otto Ludwig Hölder – Polish" lang="pl" hreflang="pl" data-title="Otto Ludwig Hölder" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Otto_H%C3%B6lder" title="Otto Hölder – Portuguese" lang="pt" hreflang="pt" data-title="Otto Hölder" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D1%91%D0%BB%D1%8C%D0%B4%D0%B5%D1%80,_%D0%9E%D1%82%D1%82%D0%BE" title="Гёльдер, Отто – Russian" lang="ru" hreflang="ru" data-title="Гёльдер, Отто" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Otto_Ludwig_H%C3%B6lder" title="Otto Ludwig Hölder – Slovak" lang="sk" hreflang="sk" data-title="Otto Ludwig Hölder" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Otto_Ludwig_H%C3%B6lder" title="Otto Ludwig Hölder – Slovenian" lang="sl" hreflang="sl" data-title="Otto Ludwig Hölder" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Otto_H%C3%B6lder" title="Otto Hölder – Swedish" lang="sv" hreflang="sv" data-title="Otto Hölder" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AD%E0%B9%87%E0%B8%AD%E0%B8%97%E0%B9%82%E0%B8%97_%E0%B9%80%E0%B8%AE%E0%B8%B4%E0%B8%A5%E0%B9%80%E0%B8%94%E0%B8%AD%E0%B8%A3%E0%B9%8C" title="อ็อทโท เฮิลเดอร์ – Thai" lang="th" hreflang="th" data-title="อ็อทโท เฮิลเดอร์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9E%D1%82%D1%82%D0%BE_%D2%B2%D3%AF%D0%BB%D0%B4%D0%B5%D1%80" title="Отто Ҳӯлдер – Tajik" lang="tg" hreflang="tg" data-title="Отто Ҳӯлдер" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Otto_H%C3%B6lder" title="Otto Hölder – Turkish" lang="tr" hreflang="tr" data-title="Otto Hölder" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9E%D1%82%D1%82%D0%BE_%D0%93%D0%B5%D0%BB%D1%8C%D0%B4%D0%B5%D1%80" title="Отто Гельдер – Ukrainian" lang="uk" hreflang="uk" data-title="Отто Гельдер" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A5%A7%E6%89%98%C2%B7%E8%B5%AB%E7%88%BE%E5%BE%B7" title="奧托·赫爾德 – Chinese" lang="zh" hreflang="zh" data-title="奧托·赫爾德" data-language-autonym="中文" data-language-local-name="Chinese" 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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">German mathematician (1859–1937)</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox biography vcard"><tbody><tr><th colspan="2" class="infobox-above" style="font-size:125%;"><div class="fn">Ludwig Otto Hölder</div></th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Hoelder_Otto.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/8/82/Hoelder_Otto.jpg" decoding="async" width="142" height="143" class="mw-file-element" data-file-width="142" data-file-height="143" /></a></span><div class="infobox-caption">Otto Hölder</div></td></tr><tr><th scope="row" class="infobox-label">Born</th><td class="infobox-data"><span style="display:none">(<span class="bday">1859-12-22</span>)</span>22 December 1859<br /><div style="display:inline" class="birthplace"><a href="/wiki/Stuttgart" title="Stuttgart">Stuttgart</a>, <a href="/wiki/Kingdom_of_W%C3%BCrttemberg" title="Kingdom of Württemberg">Kingdom of Württemberg</a></div></td></tr><tr><th scope="row" class="infobox-label">Died</th><td class="infobox-data">29 August 1937<span style="display:none">(1937-08-29)</span> (aged 77)<br /><div style="display:inline" class="deathplace"><a href="/wiki/Leipzig" title="Leipzig">Leipzig</a>, <a href="/wiki/Nazi_Germany" title="Nazi Germany">Germany</a></div></td></tr><tr><th scope="row" class="infobox-label">Nationality</th><td class="infobox-data category"><a href="/wiki/Germany" title="Germany">German</a></td></tr><tr><th scope="row" class="infobox-label">Education</th><td class="infobox-data"><a href="/wiki/University_of_Stuttgart" title="University of Stuttgart">University of Stuttgart</a><br /><a href="/wiki/University_of_Berlin" class="mw-redirect" title="University of Berlin">University of Berlin</a><br /><a href="/wiki/University_of_T%C3%BCbingen" title="University of Tübingen">University of Tübingen</a></td></tr><tr><th scope="row" class="infobox-label">Known for</th><td class="infobox-data"><a href="/wiki/H%C3%B6lder_condition" title="Hölder condition">Hölder condition</a><br /><a href="/wiki/H%C3%B6lder_mean" class="mw-redirect" title="Hölder mean">Hölder mean</a><br /><a href="/wiki/H%C3%B6lder_summation" title="Hölder summation">Hölder summation</a><br /><a href="/wiki/H%C3%B6lder%27s_inequality" title="Hölder's inequality">Hölder's inequality</a><br /><a href="/wiki/H%C3%B6lder%27s_theorem" title="Hölder's theorem">Hölder's theorem</a><br /><a href="/wiki/Composition_series" title="Composition series">Jordan–Hölder theorem</a></td></tr><tr><th scope="row" class="infobox-label">Spouse</th><td class="infobox-data">Helene Hölder</td></tr><tr><th scope="row" class="infobox-label">Children</th><td class="infobox-data"><a href="/wiki/Ernst_H%C3%B6lder" title="Ernst Hölder">Ernst Hölder</a></td></tr><tr><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546" /><b>Scientific career</b></td></tr><tr><th scope="row" class="infobox-label">Fields</th><td class="infobox-data category"><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></td></tr><tr><th scope="row" class="infobox-label">Institutions</th><td class="infobox-data"><a href="/wiki/University_of_G%C3%B6ttingen" title="University of Göttingen">University of Göttingen</a><br /><a href="/wiki/University_of_T%C3%BCbingen" title="University of Tübingen">University of Tübingen</a><br /><a href="/wiki/University_of_Leipzig" class="mw-redirect" title="University of Leipzig">University of Leipzig</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Doctoral_advisor" title="Doctoral advisor">Doctoral advisor</a></th><td class="infobox-data"><a href="/wiki/Paul_du_Bois-Reymond" title="Paul du Bois-Reymond">Paul du Bois-Reymond</a><sup id="cite_ref-mg_1-0" class="reference"><a href="#cite_note-mg-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></td></tr><tr><th scope="row" class="infobox-label">Doctoral students</th><td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist"> <ul><li><a href="/wiki/Emil_Artin" title="Emil Artin">Emil Artin</a><sup id="cite_ref-mg_1-1" class="reference"><a href="#cite_note-mg-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/William_Threlfall" title="William Threlfall">William Threlfall</a><sup id="cite_ref-mg_1-2" class="reference"><a href="#cite_note-mg-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Hermann_Vermeil" title="Hermann Vermeil">Hermann Vermeil</a><sup id="cite_ref-mg_1-3" class="reference"><a href="#cite_note-mg-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li></ul> </div></td></tr><tr style="display:none"><td colspan="2"> </td></tr></tbody></table> <p><b>Ludwig Otto Hölder</b> (December 22, 1859 – August 29, 1937) was a German <a href="/wiki/Mathematician" title="Mathematician">mathematician</a> born in <a href="/wiki/Stuttgart" title="Stuttgart">Stuttgart</a>.<sup id="cite_ref-mactutor_2-0" class="reference"><a href="#cite_note-mactutor-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Early_life_and_education">Early life and education</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Otto_H%C3%B6lder&action=edit&section=1" title="Edit section: Early life and education"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Hölder was the youngest of three sons of professor Otto Hölder (1811–1890), and a grandson of professor Christian Gottlieb Hölder (1776–1847); his two brothers also became professors. He first studied at the <i>Polytechnikum</i> (which today is the <a href="/wiki/University_of_Stuttgart" title="University of Stuttgart">University of Stuttgart</a>) and then in 1877 went to <a href="/wiki/Berlin" title="Berlin">Berlin</a> where he was a student of <a href="/wiki/Leopold_Kronecker" title="Leopold Kronecker">Leopold Kronecker</a>, <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a>, and <a href="/wiki/Ernst_Kummer" title="Ernst Kummer">Ernst Kummer</a>.<sup id="cite_ref-mactutor_2-1" class="reference"><a href="#cite_note-mactutor-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In 1877, he entered the <a href="/wiki/University_of_Berlin" class="mw-redirect" title="University of Berlin">University of Berlin</a> and took his doctorate from the <a href="/wiki/University_of_T%C3%BCbingen" title="University of Tübingen">University of Tübingen</a> in 1882. The title of his doctoral thesis was "Beiträge zur Potentialtheorie" ("Contributions to <a href="/wiki/Potential_theory" title="Potential theory">potential theory</a>").<sup id="cite_ref-mg_1-4" class="reference"><a href="#cite_note-mg-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Following this, he went to the <a href="/wiki/University_of_Leipzig" class="mw-redirect" title="University of Leipzig">University of Leipzig</a> but was unable to <a href="/wiki/Habilitation" title="Habilitation">habilitate</a> there, instead earning a second doctorate and habilitation at the <a href="/wiki/University_of_G%C3%B6ttingen" title="University of Göttingen">University of Göttingen</a>, both in 1884. </p> <div class="mw-heading mw-heading2"><h2 id="Academic_career_and_later_life">Academic career and later life</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Otto_H%C3%B6lder&action=edit&section=2" title="Edit section: Academic career and later life"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>He was unable to get government approval for a faculty position in Göttingen, and instead was offered a position as extraordinary professor at Tübingen in 1889. Temporary mental incapacitation delayed his acceptance but he began working there in 1890. In 1899, he took the former chair of <a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a> as a full professor at the University of Leipzig. There he served as dean from 1912 to 1913, and as rector in 1918.<sup id="cite_ref-mactutor_2-2" class="reference"><a href="#cite_note-mactutor-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>He married Helene, the daughter of a bank director and politician, in 1899. They had two sons and two daughters. His son <a href="/wiki/Ernst_H%C3%B6lder" title="Ernst Hölder">Ernst Hölder</a> became another mathematician,<sup id="cite_ref-mactutor_2-3" class="reference"><a href="#cite_note-mactutor-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and his daughter Irmgard married mathematician <a href="/wiki/Aurel_Wintner" title="Aurel Wintner">Aurel Wintner</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>In 1933, Hölder signed the <i><a href="/wiki/Vow_of_allegiance_of_the_Professors_of_the_German_Universities_and_High-Schools_to_Adolf_Hitler_and_the_National_Socialistic_State" title="Vow of allegiance of the Professors of the German Universities and High-Schools to Adolf Hitler and the National Socialistic State">Vow of allegiance of the Professors of the German Universities and High-Schools to Adolf Hitler and the National Socialistic State</a></i>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_contributions">Mathematical contributions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Otto_H%C3%B6lder&action=edit&section=3" title="Edit section: Mathematical contributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Holder%27s_inequality" class="mw-redirect" title="Holder's inequality">Holder's inequality</a>, named for Hölder, was actually <a href="/wiki/Mathematical_proof" title="Mathematical proof">proven</a> earlier by <a href="/wiki/Leonard_James_Rogers" title="Leonard James Rogers">Leonard James Rogers</a>. It is named for a paper in which Hölder, citing Rogers, reproves it;<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> in turn, the same paper includes a proof of what is now called <a href="/wiki/Jensen%27s_inequality" title="Jensen's inequality">Jensen's inequality</a>, with some side conditions that were later removed by Jensen.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Hölder is also noted for many other <a href="/wiki/Theorem" title="Theorem">theorems</a> including the <a href="/wiki/Jordan%E2%80%93H%C3%B6lder_theorem" class="mw-redirect" title="Jordan–Hölder theorem">Jordan–Hölder theorem</a>, the theorem stating that every <a href="/wiki/Linearly_ordered_group" title="Linearly ordered group">linearly ordered group</a> that satisfies an <a href="/wiki/Archimedean_property" title="Archimedean property">Archimedean property</a> is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of the additive <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> of <a href="/wiki/Real_number" title="Real number">real numbers</a>, the classification of <a href="/wiki/Simple_group" title="Simple group">simple groups</a> of <a href="/wiki/Order_of_a_group" class="mw-redirect" title="Order of a group">order</a> up to 200, the <a href="/wiki/Automorphisms_of_the_symmetric_and_alternating_groups#The_exceptional_outer_automorphism_of_S6" title="Automorphisms of the symmetric and alternating groups">anomalous outer automorphisms</a> of the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> <i>S</i><sub>6</sub>, and <a href="/wiki/H%C3%B6lder%27s_theorem" title="Hölder's theorem">Hölder's theorem</a>, which implies that the <a href="/wiki/Gamma_function" title="Gamma function">Gamma function</a> satisfies no <a href="/wiki/Algebraic_differential_equation" title="Algebraic differential equation">algebraic differential equation</a>. Another idea related to his name is the <a href="/wiki/H%C3%B6lder_condition" title="Hölder condition">Hölder condition</a> (or Hölder continuity), which is used in many areas of <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a>, including the theories of <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a> and <a href="/wiki/Function_space" title="Function space">function spaces</a>. </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Otto_H%C3%B6lder&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-mg-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-mg_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-mg_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-mg_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-mg_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-mg_1-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://mathgenealogy.org/id.php?id=18608">Otto Hölder</a> at the <a href="/wiki/Mathematics_Genealogy_Project" title="Mathematics Genealogy Project">Mathematics Genealogy Project</a></span> </li> <li id="cite_note-mactutor-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-mactutor_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-mactutor_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-mactutor_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-mactutor_2-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" class="mw-redirect" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Holder.html">"Otto Hölder"</a>, <i><a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a></i>, <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Otto+H%C3%B6lder&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FHolder.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOtto+H%C3%B6lder" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFElbertGaray2006" class="citation cs2">Elbert, Árpád; Garay, Barnabás M. (2006), "Differential equations: Hungary, the extended first half of the 20th century", in Horváth, János (ed.), <i>A Panorama of Hungarian Mathematics in the Twentieth Century, I</i>, Bolyai Soc. Math. Stud., vol. 14, Springer, Berlin, pp. <span class="nowrap">245–</span>294, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-30721-1_9">10.1007/978-3-540-30721-1_9</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-28945-6" title="Special:BookSources/978-3-540-28945-6"><bdi>978-3-540-28945-6</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2547513">2547513</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Differential+equations%3A+Hungary%2C+the+extended+first+half+of+the+20th+century&rft.btitle=A+Panorama+of+Hungarian+Mathematics+in+the+Twentieth+Century%2C+I&rft.series=Bolyai+Soc.+Math.+Stud.&rft.pages=%3Cspan+class%3D%22nowrap%22%3E245-%3C%2Fspan%3E294&rft.pub=Springer%2C+Berlin&rft.date=2006&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2547513%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-540-30721-1_9&rft.isbn=978-3-540-28945-6&rft.aulast=Elbert&rft.aufirst=%C3%81rp%C3%A1d&rft.au=Garay%2C+Barnab%C3%A1s+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOtto+H%C3%B6lder" class="Z3988"></span>; see <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EWm4WzSaG3IC&pg=PA248">p. 248</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://archive.org/details/bekenntnisderpro00natiuoft/page/135/mode/1up"><i>Bekenntnis der Professoren an den Universitäten und Hochschulen zu Adolf Hitler und dem nationalsozialistischen Staat; überreicht vom Nationalsozialistischen Lehrerbund Deutschland-Sachsen</i></a>, Dresden, 1933, p. 135</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bekenntnis+der+Professoren+an+den+Universit%C3%A4ten+und+Hochschulen+zu+Adolf+Hitler+und+dem+nationalsozialistischen+Staat%3B+%C3%BCberreicht+vom+Nationalsozialistischen+Lehrerbund+Deutschland-Sachsen&rft.pages=135&rft.pub=Dresden&rft.date=1933&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbekenntnisderpro00natiuoft%2Fpage%2F135%2Fmode%2F1up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOtto+H%C3%B6lder" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMaligranda1998" class="citation cs2">Maligranda, Lech (1998), "Why Hölder's inequality should be called Rogers' inequality", <i>Mathematical Inequalities & Applications</i>, <b>1</b> (1): <span class="nowrap">69–</span>83, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.7153%2Fmia-01-05">10.7153/mia-01-05</a></span>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1492911">1492911</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Inequalities+%26+Applications&rft.atitle=Why+H%C3%B6lder%27s+inequality+should+be+called+Rogers%27+inequality&rft.volume=1&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E69-%3C%2Fspan%3E83&rft.date=1998&rft_id=info%3Adoi%2F10.7153%2Fmia-01-05&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1492911%23id-name%3DMR&rft.aulast=Maligranda&rft.aufirst=Lech&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOtto+H%C3%B6lder" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGuessabSchmeisser2013" class="citation cs2">Guessab, A.; Schmeisser, G. (2013), "Necessary and sufficient conditions for the validity of Jensen's inequality", <i><a href="/wiki/Archiv_der_Mathematik" title="Archiv der Mathematik">Archiv der Mathematik</a></i>, <b>100</b> (6): <span class="nowrap">561–</span>570, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00013-013-0522-3">10.1007/s00013-013-0522-3</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3069109">3069109</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:56372266">56372266</a>, <q>under the additional assumption that <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 \varphi ''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>φ</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce142194a489f1fcf6249b52ca9a3bd79bafa289" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.657ex; height:3.009ex;" alt="{\displaystyle \varphi ''}" /></span> exists, this inequality was already obtained by Hölder in 1889</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archiv+der+Mathematik&rft.atitle=Necessary+and+sufficient+conditions+for+the+validity+of+Jensen%27s+inequality&rft.volume=100&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E561-%3C%2Fspan%3E570&rft.date=2013&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3069109%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A56372266%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00013-013-0522-3&rft.aulast=Guessab&rft.aufirst=A.&rft.au=Schmeisser%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOtto+H%C3%B6lder" class="Z3988"></span></span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist 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