{"id":"https://openalex.org/W3205889583","doi":"https://doi.org/10.3390/fractalfract5040172","title":"(k, \u03c8)-Proportional Fractional Integral P\u00f3lya\u2013Szeg\u00f6- and Gr\u00fcss-Type Inequalities","display_name":"(k, \u03c8)-Proportional Fractional Integral P\u00f3lya\u2013Szeg\u00f6- and Gr\u00fcss-Type Inequalities","publication_year":2021,"publication_date":"2021-10-18","ids":{"openalex":"https://openalex.org/W3205889583","doi":"https://doi.org/10.3390/fractalfract5040172","mag":"3205889583"},"language":"en","primary_location":{"id":"doi:10.3390/fractalfract5040172","doi":"https://doi.org/10.3390/fractalfract5040172","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/fractalfract5040172","pdf_url":"https://www.mdpi.com/2504-3110/5/4/172/pdf?version=1638771064","source":{"id":"https://openalex.org/S4210227020","display_name":"Fractal and Fractional","issn_l":"2504-3110","issn":["2504-3110"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","norway-1","poland-20","russia-white-list-1"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Fractal and Fractional","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2504-3110/5/4/172/pdf?version=1638771064","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5069991422","display_name":"Tariq A. Aljaaidi","orcid":"https://orcid.org/0000-0003-4316-5895","observed_orcids":["https://orcid.org/0000-0003-4316-5895"]},"institutions":[{"id":"https://openalex.org/I165259314","display_name":"Dr. Babasaheb Ambedkar Marathwada University","ror":"https://ror.org/033pfj584","country_code":"IN","type":"education","lineage":["https://openalex.org/I165259314"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Tariq A. Aljaaidi","raw_affiliation_strings":["Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431001, Maharashtra, India"],"raw_orcid":"https://orcid.org/0000-0003-4316-5895","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431001, Maharashtra, India","institution_ids":["https://openalex.org/I165259314"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5046821818","display_name":"Deepak B. Pachpatte","orcid":"https://orcid.org/0000-0003-3763-4878","observed_orcids":["https://orcid.org/0000-0003-3763-4878"]},"institutions":[{"id":"https://openalex.org/I165259314","display_name":"Dr. Babasaheb Ambedkar Marathwada University","ror":"https://ror.org/033pfj584","country_code":"IN","type":"education","lineage":["https://openalex.org/I165259314"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Deepak B. Pachpatte","raw_affiliation_strings":["Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431001, Maharashtra, India"],"raw_orcid":"https://orcid.org/0000-0003-3763-4878","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431001, Maharashtra, India","institution_ids":["https://openalex.org/I165259314"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5070524846","display_name":"Mohammed S. \u202cAbdo","orcid":"https://orcid.org/0000-0001-9085-324X","observed_orcids":["https://orcid.org/0000-0001-9085-324X"]},"institutions":[{"id":"https://openalex.org/I31605765","display_name":"Hodeidah University","ror":"https://ror.org/05fkpm735","country_code":"YE","type":"education","lineage":["https://openalex.org/I31605765"]}],"countries":["YE"],"is_corresponding":false,"raw_author_name":"Mohammed S. Abdo","raw_affiliation_strings":["Department of Mathematics, Hodeidah University, Al-Hodeidah, Yemen"],"raw_orcid":"https://orcid.org/0000-0001-9085-324X","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Hodeidah University, Al-Hodeidah, Yemen","institution_ids":["https://openalex.org/I31605765"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5078096818","display_name":"Thongchai Botmart","orcid":"https://orcid.org/0000-0001-9325-5560","observed_orcids":["https://orcid.org/0000-0001-9325-5560"]},"institutions":[{"id":"https://openalex.org/I179193067","display_name":"Khon Kaen University","ror":"https://ror.org/03cq4gr50","country_code":"TH","type":"education","lineage":["https://openalex.org/I179193067"]}],"countries":["TH"],"is_corresponding":false,"raw_author_name":"Thongchai Botmart","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"],"raw_orcid":"https://orcid.org/0000-0001-9325-5560","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand","institution_ids":["https://openalex.org/I179193067"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5114376646","display_name":"Hijaz Ahmad","orcid":"https://orcid.org/0000-0002-5438-5407","observed_orcids":["https://orcid.org/0000-0002-5438-5407"]},"institutions":[{"id":"https://openalex.org/I4210155399","display_name":"UniNettuno University","ror":"https://ror.org/04q0nep37","country_code":"IT","type":"education","lineage":["https://openalex.org/I4210155399"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Hijaz Ahmad","raw_affiliation_strings":["Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy"],"raw_orcid":"https://orcid.org/0000-0002-5438-5407","affiliations":[{"raw_affiliation_string":"Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy","institution_ids":["https://openalex.org/I4210155399"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5020974388","display_name":"Mohammed A. \u202cAlmalahi","orcid":"https://orcid.org/0000-0001-5719-086X","observed_orcids":["https://orcid.org/0000-0001-5719-086X"]},"institutions":[{"id":"https://openalex.org/I4405254395","display_name":"Hajjah University","ror":"https://ror.org/01rpcwa78","country_code":"YE","type":"education","lineage":["https://openalex.org/I4405254395"]}],"countries":["YE"],"is_corresponding":false,"raw_author_name":"Mohammed A. Almalahi","raw_affiliation_strings":["Department of Mathematics, Hajjah University, Hajjah, Yemen"],"raw_orcid":"https://orcid.org/0000-0001-5719-086X","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Hajjah University, Hajjah, Yemen","institution_ids":["https://openalex.org/I4405254395"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5036693928","display_name":"Saleh S. Redhwan","orcid":"https://orcid.org/0000-0002-4779-7657","observed_orcids":["https://orcid.org/0000-0002-4779-7657"]},"institutions":[{"id":"https://openalex.org/I165259314","display_name":"Dr. Babasaheb Ambedkar Marathwada University","ror":"https://ror.org/033pfj584","country_code":"IN","type":"education","lineage":["https://openalex.org/I165259314"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Saleh S. Redhwan","raw_affiliation_strings":["Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431001, Maharashtra, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431001, Maharashtra, India","institution_ids":["https://openalex.org/I165259314"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5114376646"],"corresponding_institution_ids":["https://openalex.org/I4210155399"],"apc_list":{"value":1600,"currency":"CHF","value_usd":1728},"apc_paid":{"value":1600,"currency":"CHF","value_usd":1728},"fwci":2.507,"has_fulltext":true,"cited_by_count":17,"citation_normalized_percentile":{"value":0.88993904,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":98},"biblio":{"volume":"5","issue":"4","first_page":"172","last_page":"172"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12062","display_name":"Approximation Theory and Sequence Spaces","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/polya-szego-inequality","display_name":"P\u00f3lya-Szeg\u0151 inequality","score":0.9179999828338623},{"id":"https://openalex.org/keywords/gruss-type-inequalities","display_name":"Gr\u00fcss type inequalities","score":0.9010000228881836},{"id":"https://openalex.org/keywords/k-fractional-integrals","display_name":"k-fractional integrals","score":0.8349999785423279},{"id":"https://openalex.org/keywords/generalized-fractional-operators","display_name":"generalized fractional operators","score":0.6240000128746033},{"id":"https://openalex.org/keywords/integral-inequalities","display_name":"integral inequalities","score":0.5249999761581421}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7777312994003296},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.6612218618392944},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5507199168205261},{"id":"https://openalex.org/C45555294","wikidata":"https://www.wikidata.org/wiki/Q28113351","display_name":"Inequality","level":2,"score":0.5407547950744629},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.49990057945251465},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.47201913595199585},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4649542570114136},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.36683160066604614},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/fractalfract5040172","doi":"https://doi.org/10.3390/fractalfract5040172","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/fractalfract5040172","pdf_url":"https://www.mdpi.com/2504-3110/5/4/172/pdf?version=1638771064","source":{"id":"https://openalex.org/S4210227020","display_name":"Fractal and Fractional","issn_l":"2504-3110","issn":["2504-3110"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","norway-1","poland-20","russia-white-list-1"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Fractal and Fractional","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:08023aa24aa746a6ae1cf3deb1fd26ca","doi":null,"is_oa":true,"landing_page_url":"https://doaj.org/article/08023aa24aa746a6ae1cf3deb1fd26ca","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Fractal and Fractional, Vol 5, Iss 4, p 172 (2021)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/fractalfract5040172","doi":"https://doi.org/10.3390/fractalfract5040172","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/fractalfract5040172","pdf_url":"https://www.mdpi.com/2504-3110/5/4/172/pdf?version=1638771064","source":{"id":"https://openalex.org/S4210227020","display_name":"Fractal and Fractional","issn_l":"2504-3110","issn":["2504-3110"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","norway-1","poland-20","russia-white-list-1"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Fractal and Fractional","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3205889583.pdf","grobid_xml":"https://content.openalex.org/works/W3205889583.grobid-xml"},"referenced_works_count":39,"referenced_works":["https://openalex.org/W633640090","https://openalex.org/W1971200514","https://openalex.org/W2006182851","https://openalex.org/W2035906688","https://openalex.org/W2041447043","https://openalex.org/W2052082751","https://openalex.org/W2057896742","https://openalex.org/W2083432344","https://openalex.org/W2087101294","https://openalex.org/W2276336683","https://openalex.org/W2340261695","https://openalex.org/W2380581110","https://openalex.org/W2382065857","https://openalex.org/W2473195096","https://openalex.org/W2488882192","https://openalex.org/W2510366261","https://openalex.org/W2513201624","https://openalex.org/W2531454945","https://openalex.org/W2757679221","https://openalex.org/W2790996674","https://openalex.org/W2805614445","https://openalex.org/W2883444645","https://openalex.org/W2898923932","https://openalex.org/W2949897179","https://openalex.org/W2951682740","https://openalex.org/W2960490161","https://openalex.org/W2972177571","https://openalex.org/W2981313766","https://openalex.org/W2991368749","https://openalex.org/W2999914773","https://openalex.org/W3000109662","https://openalex.org/W3088317680","https://openalex.org/W3096363020","https://openalex.org/W3108056518","https://openalex.org/W3129236547","https://openalex.org/W3156335426","https://openalex.org/W4205333027","https://openalex.org/W4247371619","https://openalex.org/W4247953052"],"related_works":["https://openalex.org/W3028969026","https://openalex.org/W2340261695","https://openalex.org/W4400952960","https://openalex.org/W3033737287","https://openalex.org/W3030977879","https://openalex.org/W4411065124","https://openalex.org/W3007018644","https://openalex.org/W2993810348","https://openalex.org/W4210489240","https://openalex.org/W4293104791"],"abstract_inverted_index":{"The":[0,79],"purpose":[1],"of":[2,16,31,57,66,110],"this":[3,67,83,117],"research":[4],"was":[5],"to":[6,11,24,44,91],"discover":[7],"a":[8,100],"novel":[9],"method":[10],"recover":[12],"k-fractional":[13,96],"integral":[14,29,77,97],"inequalities":[15,23,30,98,115],"the":[17,32,38,64,111],"P\u00f3lya\u2013Szeg\u00f6-type.":[18],"We":[19],"employ":[20],"these":[21],"generalized":[22,68],"investigate":[25,71],"some":[26,56,72,106],"new":[27,101,107],"fractional":[28,40,76],"Gr\u00fcss-type.":[33],"More":[34],"precisely,":[35],"we":[36,52,70,104],"generalize":[37],"proportional":[39],"operators":[41],"with":[42],"respect":[43],"another":[45],"strictly":[46],"increasing":[47],"continuous":[48],"function":[49],"\u03c8.":[50],"Then,":[51],"state":[53],"and":[54,60,74,94,113],"prove":[55],"its":[58],"properties":[59],"special":[61,108],"cases.":[62],"With":[63],"help":[65],"operator,":[69],"P\u00f3lya\u2013Szeg\u00f6-":[73,93,112],"Gr\u00fcss-type":[75,95,114],"inequalities.":[78],"functions":[80,90],"used":[81],"in":[82,99],"work":[84],"are":[85],"bounded":[86],"by":[87],"two":[88],"positive":[89],"obtain":[92],"sense.":[102],"Moreover,":[103],"discuss":[105],"cases":[109],"through":[116],"work.":[118]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":6},{"year":2022,"cited_by_count":3}],"updated_date":"2026-11-10T11:39:43.021295","created_date":"2021-10-25T00:00:00"}
