{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:16:37Z","timestamp":1760058997091,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,15]],"date-time":"2025-05-15T00:00:00Z","timestamp":1747267200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper concerns the old problem of the connection between the dilatations of a given quasisymmetric homeomorphism h of a circle and the associated polygonal quasiconformal maps with a fixed finite number of boundary points, namely whether k(h)=supkn, where the supremum is taken over all possible n-gons formed by the disk with n distinguished boundary points. A still open question is whether such equality is valid under the additional assumption that the naturally related univalent functions with quasiconformal extensions have equal Grunsky and Teichm\u00fcller norms. We solved this problem in the negative for n\u22654.<\/jats:p>","DOI":"10.3390\/axioms14050372","type":"journal-article","created":{"date-parts":[[2025,5,15]],"date-time":"2025-05-15T06:59:16Z","timestamp":1747292356000},"page":"372","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Polygonal Quasiconformality and Grunsky\u2019s Operator"],"prefix":"10.3390","volume":"14","author":[{"given":"Samuel L.","family":"Krushkal","sequence":"first","affiliation":[{"name":"Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel"},{"name":"Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,15]]},"reference":[{"unstructured":"Ahlfors, L.V. (1966). Lectures on Quasiconformal Mappings, Van Nostrand.","key":"ref_1"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1007\/BF02566018","article-title":"Quadrilaterals and extremal quasiconformal extensions","volume":"70","author":"Anderson","year":"1995","journal-title":"Comment. Math. Helv."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1479","DOI":"10.1090\/S0002-9939-97-03863-X","article-title":"An approximation condition and extremal quasiconformal extensions","volume":"125","author":"Reich","year":"1997","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1007\/s000140050036","article-title":"Moduli of quadrilaterals and extremal quasiconformal extensions of quasisymmetric functions","volume":"72","author":"Wu","year":"1997","journal-title":"Comment. Math. Helv."},{"key":"ref_5","first-page":"141","article-title":"Einige neuere Entwicklungen bei quasikonformen Abbildungen","volume":"94","year":"1992","journal-title":"Jber. Deutsch. Math. Verein."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1007\/s000140050080","article-title":"On the dilatation of extremal quasiconformal mappings of polygons","volume":"74","author":"Strebel","year":"1999","journal-title":"Comment. Math. Helv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/BF02786555","article-title":"Polygonal quasiconformal maps and Grunsky inequalities","volume":"90","author":"Krushkal","year":"2003","journal-title":"J. Anal. Math."},{"key":"ref_8","first-page":"604","article-title":"The Grunsky operator and quasiconformality: Old and new","volume":"288","author":"Krushkal","year":"2025","journal-title":"J. Math. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1016\/S0377-0427(99)00017-5","article-title":"Remarks on the Grunsky norm and pth root transformation","volume":"105","author":"Grinshpan","year":"1999","journal-title":"J. Comput. Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/BF01580272","article-title":"Koeffizientenbedingungen f\u00fcr schlicht abbildende meromorphe Funktionen","volume":"45","author":"Grunsky","year":"1939","journal-title":"Math. Z."},{"unstructured":"Krushkal, S.L., and K\u00fchnau, R. (1983). Quasikonforme Abbildungen\u2014Neue Methode und Anwendungen, Teubner-Texte zur Math., Bd. 54, Teubner.","key":"ref_11"},{"unstructured":"Pommerenke, C. (1975). Univalent Functions, Vandenhoeck & Ruprecht.","key":"ref_12"},{"unstructured":"Zhuravlev, I.V. (1979). Univalent Functions and Teichm\u00fcller Spaces, Inst. of Mathematics. (In Russian).","key":"ref_13"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/s10958-024-06988-3","article-title":"Analytic and geometric quasiinvariants of convex curvelinear polygons with infinite number of vertices","volume":"279","author":"Krushkal","year":"2024","journal-title":"J. Math. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"650","DOI":"10.1007\/BF02564699","article-title":"Grunsky coefficient inequalities, Carath\u00e9odory metric and extremal quasiconformal mappings","volume":"64","author":"Krushkal","year":"1989","journal-title":"Comment. Math. Helv."},{"unstructured":"Ahlfors, L.V., Kra, I., Maskit, B., and Nirenberg, L. (1974). Extremal Quasiconformal Mappings with Given Boundary Values, Academic Press. Contributions to Analysis.","key":"ref_16"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/372\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:33:10Z","timestamp":1760031190000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/372"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,15]]},"references-count":16,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["axioms14050372"],"URL":"https:\/\/doi.org\/10.3390\/axioms14050372","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,5,15]]}}}