{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T03:20:27Z","timestamp":1768706427128,"version":"3.49.0"},"reference-count":40,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,2,21]],"date-time":"2021-02-21T00:00:00Z","timestamp":1613865600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work we establish a few equivalent statements of a Hilbert-type integral inequality in the whole plane related to the kernel of the arc tangent function. We prove that the constant factor, which is associated with the cosine function, is optimal. Some special cases as well as some operator expressions are also presented.<\/jats:p>","DOI":"10.3390\/sym13020351","type":"journal-article","created":{"date-parts":[[2021,2,21]],"date-time":"2021-02-21T23:59:31Z","timestamp":1613951971000},"page":"351","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Hilbert-Type Integral Inequality in the Whole Plane Related to the Arc Tangent Function"],"prefix":"10.3390","volume":"13","author":[{"given":"Michael Th.","family":"Rassias","sequence":"first","affiliation":[{"name":"Institute of Mathematics, University of Zurich, CH-8057 Zurich, Switzerland"},{"name":"Program in Interdisciplinary Studies, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bicheng","family":"Yang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Guangdong University of Education, Guangzhou 510303, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andrei","family":"Raigorodskii","sequence":"additional","affiliation":[{"name":"Moscow Institute of Physics and Technology, Institutskiy per, d. 9, 141700 Dolgoprudny, Russia"},{"name":"Faculty of Bioengineering and Bioinformatics, Moscow State University, 119991 Moscow, Russia"},{"name":"Institute of Mathematics and Computer Science, Buryat State University, 670000 Ulan-Ude, Russia"},{"name":"Caucasus Mathematical Center, Adyghe State University, 352700 Maykop, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,2,21]]},"reference":[{"key":"ref_1","unstructured":"Hardy, G.H., Littlewood, J.E., and P\u00f3lya, G. 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