Document JFM 44.0759.03

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Nachr.</a> <a href="/?q=in%3A320442" title="Articles in this Issue">1913, 286-288 (1913)</a>. </div> <div class="abstract">“Die Frage der R盲nderzuordnung bei konformer Abbildung eines schlichten einfach zusammenh盲ngenden Bereichs $ B $ in der $f$- Ebene auf die Fl盲che des Einheitskreises $ E $ einer $z$-Ebene ist neuerdings von <span class="zbmathjax-textit">Carath茅odory</span> (Math. Ann. 73, 305-320, 323-370) und <span class="zbmathjax-textit">Study</span> ( “Vorlesungen 眉ber Geometrie” Bd. II, Leipzig, Teubner 1913, insbesondere $\S$ 5-8) behandelt worden, indem diese Autoren einerseits f眉r den Fall, da脽<span class="zbmathjax-space"></span>die Begrenzung eine <span class="zbmathjax-textit">Jordan</span>sche Kurve ist, eine <span class="zbmathjax-textit">Osgood</span>sche Vermutung best盲tigten, da脽<span class="zbmathjax-space"></span>n盲mlich die konforme Abbildung sich auch noch auf den Rand selbst als stetige Abbildung fortsetzt, andererseits im Falle des schlechthin allgemeinsten Bereiches den Satz bewiesen, da脽<span class="zbmathjax-space"></span>die Menge $ M $ der vom Innern des Bereichs $ B $ her erreichbaren Randpunkte von $ B, $ welche offenbar zyklisch geordnet ist, eineindeutig und dem Anordnungstypus entsprechend auf eine Menge $ M' $ von Peripheriepunkten des Einheitskreises bezogen wird, welch letztere Menge auf der Peripherie 眉berall dicht verteilt ist, soda脽<span class="zbmathjax-space"></span>kein noch so kleines Peripheriest眉ck von Punkten der Menge $ M' $ frei ist. Durch den letzteren Satz wird dann, wie eine von <span class="zbmathjax-textit">Study</span> und <span class="zbmathjax-textit">Carath茅odory</span> gegebene Analyse des Randes zeigt, das Problem der R盲nderzuordnung auch f眉r den allgemeinsten Bereich in gewisser Weise vollkommen beherrscht. (Auch diese weitergehenden S盲tze von <span class="zbmathjax-textit">Study</span> und <span class="zbmathjax-textit">Carath茅odory</span> sind, worauf ich inzwischen aufmerksam geworden bin, wesentlich bereits 1903 durch <span class="zbmathjax-textit">Osgood</span> gefunden worden. Vgl. dessen vorl盲ufige Note im Amer. Math. Soc. Bull. (2) 9, 233-235, 1903: “On the transformation of the boundary in the case of conformal mapping” Vgl. ferner <span class="zbmathjax-textit">Osgood-Taylor :</span> “Conformal transformations on the boundaries of their regions of definition”)<br class="zbmathjax-paragraph"><span class="zbmathjax-textit">Carath茅odory</span> und <span class="zbmathjax-textit">Study</span> bedienten sich zu dem Zwecke des Nachweises wesentlich eines von <span class="zbmathjax-textit">Fatou</span> gefundenen Satzes 眉ber Funktionen $\varphi(z),$ welche innerhalb des Einheitskreises $|z| <1$ regul盲r sind und dem absoluten Betrage nach unterhalb einer endlichen Schranke bleiben. Dieser Satz, welchen <span class="zbmathjax-textit">Fatou</span> aus gewissen, von <span class="zbmathjax-textit">Lebesgue</span> mit seinem eigent眉mlichen Integralbegriff gewonnenen Ergebnissen folgerte, besagt, auf die Abbildungsfunktion $f(z)$ angewandt, direkt, da脽<span class="zbmathjax-space"></span>auf der Peripherie des Einheitskreises eine Gesamtheit $M''$ von 眉berall dicht verteilten Punkten existiert von der Beschaffenheit, da脽<span class="zbmathjax-space"></span>den nach diesen Punkten vom Zentrum hinf眉hrenden Radien in $B$ Linien entsprechen, die je gegen einen bestimmten erreichbaren Randpunkt von $B$ konvergieren.<br class="zbmathjax-paragraph"><span class="zbmathjax-textit">Carath茅odory</span> spricht in seinem erstgenannten Aufsatze nun die Meinung aus, da脽<span class="zbmathjax-space"></span>es ohne das <span class="zbmathjax-textit">Lebesgue</span> sche Integral vielleicht (M. A. 307) nicht m枚glich sein w眉rde, die erw盲hnten Ergebnisse der Abbildungstheorie zu erhalten. Ich werde jedoch jetzt zeigen, da脽<span class="zbmathjax-space"></span>dies in sehr einfacher Weise m枚glich ist, soda脽<span class="zbmathjax-space"></span>die Heranziehung der <span class="zbmathjax-textit">Lebesgue</span> schen Theorie tats盲chlich als eine bedeutende Verkomplizierung erscheint.<br class="zbmathjax-paragraph">Zu dem Zwecke verallgemeinere ich einen Satz von <span class="zbmathjax-textit">Schwarz</span>.<br class="zbmathjax-paragraph">Satz von <span class="zbmathjax-textit">Schwarz</span> : Es sei $\varphi(z)$ eine analytische Funktion von $z,$ regul盲r oberhalb einer gewissen Strecke $a$ der Achse des Reellen, welche ferner die Eigenschaft hat, da脽<span class="zbmathjax-space"></span>ihre Werte bei Ann盲herung an die Strecke $a$ gleichm盲脽ig sich einem konstanten Wert n盲hern. Alsdann verh盲lt sich die Funktion $\varphi(z)$ auch noch auf der Strecke $a$ selbst regul盲r, und es ist die Funktion $\varphi(z)$ 眉berhaupt eine Konstante.<br class="zbmathjax-paragraph">Erweiterung des <span class="zbmathjax-textit">Schwarz</span> schen Satzes: Die Erweiterung besteht in folgender Modifikation der Voraussetzungen: Es wird nur verlangt, da脽<span class="zbmathjax-space"></span>man in beliebiger N盲he der Strecke $a$ noch eine l盲ngs $a$ sich hinziehende Linie $\sigma$ finden kann, auf welcher die Werte der Funktion $\varphi(z)$ sich von einer festen Konstante beliebig wenig unterscheiden, und da脽<span class="zbmathjax-space"></span>au脽erdem die Funktion in dem betrachteten Bezirke beschr盲nkt sei. Unter diesen weiteren Voraussetzungen bleibt die Behauptung bestehen.”<br class="zbmathjax-paragraph">Die ausf眉hrliche Entwickliang seiner Methode gibt <span class="zbmathjax-textit">Koebe</span> im J. f眉r Math. 145 (1915).<div class="reviewer"> Reviewer: <a href="/authors/?q=Koebe">Koebe, Prof. (Jena)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A2622769">Cited in <strong>2</strong> Documents</a></div></div> <div class="keywords"> <h3>JFM Section:</h3><a href="/?q=ut%3A%22Neunter+Abschnitt.+Analytische+Geometrie.+Kapitel+5.+Verwandtschaft%2C+Transformationen%2C+Abbildungen.+B.+Konforme+Abbildungen+und+dergleichen."">Neunter Abschnitt. Analytische Geometrie. Kapitel 5. Verwandtschaft, Transformationen, Abbildungen. B. Konforme Abbildungen und dergleichen.</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="JFM" data-itemname="JFM 44.0759.03" data-ciurl="/ci/02622769" data-biburl="/bibtex/02622769.bib" data-amsurl="/amsrefs/02622769.bib" data-xmlurl="/xml/02622769.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/02622769.pdf" title="JFM 44.0759.03 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://eudml.org/doc/58874" title="Open Access at EuDML">EuDML</a> </div> <div class="sfx" style="float: right;"> </div> </div> </article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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