{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:42:13Z","timestamp":1740134533977,"version":"3.37.3"},"reference-count":9,"publisher":"Wiley","license":[{"start":{"date-parts":[[2018,1,1]],"date-time":"2018-01-01T00:00:00Z","timestamp":1514764800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Nakhon Phanom University","award":["580010"],"award-info":[{"award-number":["580010"]}]},{"DOI":"10.13039\/501100004704","name":"National Research Council of Thailand","doi-asserted-by":"publisher","award":["580010"],"award-info":[{"award-number":["580010"]}],"id":[{"id":"10.13039\/501100004704","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004071","name":"Khon Kaen University","doi-asserted-by":"publisher","award":["580010"],"award-info":[{"award-number":["580010"]}],"id":[{"id":"10.13039\/501100004071","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2018]]},"abstract":"<jats:p>The polar derivative of a polynomial <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>p<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> of degree <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:math> with respect to a complex number <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>\u03b1<\/mml:mi><\/mml:mrow><\/mml:math> is a polynomial <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi>n<\/mml:mi><mml:mi>p<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>+<\/mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>\u03b1<\/mml:mi><mml:mo>-<\/mml:mo><mml:mi>z<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:msup><mml:mrow><mml:mi>p<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u2032<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>, denoted by <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:msub><mml:mrow><mml:mi>D<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u03b1<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mi>p<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>. Let <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mn mathvariant=\"normal\">1<\/mml:mn><mml:mo>\u2264<\/mml:mo><mml:mi>R<\/mml:mi><mml:mo>\u2264<\/mml:mo><mml:mi>k<\/mml:mi><\/mml:math>. For a polynomial <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mi>p<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> of degree <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:math> having all its zeros in <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi>z<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>\u2264<\/mml:mo><mml:mi>k<\/mml:mi><\/mml:math>, we investigate a lower bound of modulus of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:msub><mml:mrow><mml:mi>D<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u03b1<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mi>p<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> on <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi>z<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mi>R<\/mml:mi><\/mml:math>. Furthermore, we present an upper bound of modulus of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:msub><mml:mrow><mml:mi>D<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u03b1<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mi>p<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> on <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi>z<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mi>R<\/mml:mi><\/mml:math> for a polynomial <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mi>p<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> of degree <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:math> having no zero in <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi>z<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>&lt;<\/mml:mo><mml:mi>k<\/mml:mi><\/mml:math>. In particular, our results in case <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M17\"><mml:mi>R<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:math> generalize some well-known inequalities.<\/jats:p>","DOI":"10.1155\/2018\/5034607","type":"journal-article","created":{"date-parts":[[2018,3,1]],"date-time":"2018-03-01T18:31:59Z","timestamp":1519929119000},"page":"1-4","source":"Crossref","is-referenced-by-count":1,"title":["Some Bounds for the Polar Derivative of a Polynomial"],"prefix":"10.1155","volume":"2018","author":[{"given":"Jiraphorn","family":"Somsuwan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Nakhon Phanom University, Nakhon Phanom 48000, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7550-7744","authenticated-orcid":true,"given":"Keaitsuda Maneeruk","family":"Nakprasit","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"8","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1941-07510-5"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1944-08177-9"},{"key":"9","first-page":"89","volume":"7","year":"1939","journal-title":"Compositio Mathematica"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1973-0325932-8"},{"issue":"2","key":"4","first-page":"203","volume":"2","year":"1999","journal-title":"Mathematical Inequalities and Applications"},{"issue":"2","key":"1","first-page":"231","volume":"1","year":"1998","journal-title":"Mathematical Inequalities and Applications"},{"issue":"4, article 119","key":"5","volume":"9","year":"2008","journal-title":"Journal of Inequalities in Pure and Applied Mathematics"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2008.10.056"},{"volume":"11","year":"1978","series-title":"Graduate Texts in Mathematics","key":"2"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2018\/5034607.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2018\/5034607.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2018\/5034607.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2018,3,1]],"date-time":"2018-03-01T18:32:02Z","timestamp":1519929122000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/ijmms\/2018\/5034607\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018]]},"references-count":9,"alternative-id":["5034607","5034607"],"URL":"https:\/\/doi.org\/10.1155\/2018\/5034607","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2018]]}}}