{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:10:33Z","timestamp":1760127033740,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,5]],"date-time":"2023-05-05T00:00:00Z","timestamp":1683244800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science, Research and Innovation Fund (NSRF)","award":["KMUTNB-FF-66-11"],"award-info":[{"award-number":["KMUTNB-FF-66-11"]}]},{"DOI":"10.13039\/501100007345","name":"King Mongkut\u2019s University of Technology North Bangkok","doi-asserted-by":"publisher","award":["KMUTNB-FF-66-11"],"award-info":[{"award-number":["KMUTNB-FF-66-11"]}],"id":[{"id":"10.13039\/501100007345","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The term convexity and theory of inequalities is an enormous and intriguing domain of research in the realm of mathematical comprehension. Due to its applications in multiple areas of science, the theory of convexity and inequalities have recently attracted a lot of attention from historians and modern researchers. This article explores the concept of a new group of modified harmonic exponential s-convex functions. Some of its significant algebraic properties are elegantly elaborated to maintain the newly described idea. A new sort of Hermite\u2013Hadamard-type integral inequality using this new concept of the function is investigated. In addition, several new estimates of Hermite\u2013Hadamard inequality are presented to improve the study. These new results illustrate some generalizations of prior findings in the literature.<\/jats:p>","DOI":"10.3390\/axioms12050454","type":"journal-article","created":{"date-parts":[[2023,5,5]],"date-time":"2023-05-05T03:57:31Z","timestamp":1683259051000},"page":"454","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Some Hadamard-Type Integral Inequalities Involving Modified Harmonic Exponential Type Convexity"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3084-922X","authenticated-orcid":false,"given":"Asif Ali","family":"Shaikh","sequence":"first","affiliation":[{"name":"Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan"},{"name":"Department of Mathematics, Faculty of Arts and Sciences Near East University, Mersin 99138, Turkey"}]},{"given":"Evren","family":"Hincal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Arts and Sciences Near East University, Mersin 99138, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7695-2118","authenticated-orcid":false,"given":"Sotiris K.","family":"Ntouyas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Sciences, University of Ioannina, 45110 Ioannina, Greece"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8185-3539","authenticated-orcid":false,"given":"Jessada","family":"Tariboon","sequence":"additional","affiliation":[{"name":"Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8372-2532","authenticated-orcid":false,"given":"Muhammad","family":"Tariq","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Niculescu, C.P., and Persson, L.E. (2006). Convex Functions and Their Applications, Springer.","DOI":"10.1007\/0-387-31077-0"},{"key":"ref_2","first-page":"2","article-title":"On some inequalities for s\u2013convex functions and applications","volume":"333","author":"Yildiz","year":"2013","journal-title":"J. Inequal. Appl."},{"key":"ref_3","first-page":"6615948","article-title":"Novel refinements via n-polynomial harmonically s-type convex functions and Applications in special functions","volume":"2021","author":"Butt","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"9705","DOI":"10.3934\/math.2021565","article-title":"Dynamical significance of generalized fractional integral inequalities via convexity","volume":"6","author":"Ali","year":"2021","journal-title":"AIMS Math."},{"key":"ref_5","first-page":"980438","article-title":"Some integral inequalities of Hermite\u2013Hadamard type for convex functions with applications to means","volume":"2012","author":"Xi","year":"2012","journal-title":"J. Funct. Spaces"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"507560","DOI":"10.1155\/2010\/507560","article-title":"The Hermite\u2013Hadamard type inequality of GA\u2013convex functions and its applications","volume":"2010","author":"Zhang","year":"2010","journal-title":"J. Inequal. Appl."},{"key":"ref_7","first-page":"59","article-title":"New Hermite\u2013Hadamard type inequalities via p\u2013harmonic exponential type convexity and applications","volume":"4","author":"Tariq","year":"2021","journal-title":"Univ. J. Math. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Sahoo, S.K., Tariq, M., Ahmad, H., Nasir, J., Aydi, H., and Mukheimer, A. (2021). New Ostrowski-type fractional integral inequalities via generalized exponential-type convex functions and applications. Symmetry, 13.","DOI":"10.3390\/sym13081429"},{"key":"ref_9","first-page":"2011","article-title":"Exponential convexity of Petrovic and related functional","volume":"89","author":"Butt","year":"2011","journal-title":"J. Inequal. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"6874","DOI":"10.3934\/math.2020441","article-title":"New Hermite-Hadamard type inequalities for exponentially convex functions and applications","volume":"5","author":"Zhou","year":"2020","journal-title":"AIMS Math."},{"key":"ref_11","first-page":"1","article-title":"Some new Hermite\u2013Hadamard integral inequalities for convex functions","volume":"1","author":"Omotoyinbo","year":"2014","journal-title":"Int. J. Sci. Innov. Tech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2021","DOI":"10.48185\/jmam.v2i2.216","article-title":"A note on some Ostrowski-type inequalities via generalized exponentially convex function","volume":"2","author":"Tariq","year":"2021","journal-title":"J. Math. Anal. Model."},{"key":"ref_13","first-page":"5533491","article-title":"Hermite\u2013Hadamard type inequalities via generalized harmonic exponential convexity","volume":"2021","author":"Butt","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"901","DOI":"10.1007\/s10483-007-0707-z","article-title":"Weighted Chebysev\u2013Ostrowski type inequalities","volume":"28","author":"Rafiq","year":"2007","journal-title":"Appl. Math. Mech."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1414","DOI":"10.1515\/math-2017-0121","article-title":"Some new inequalities of Hermite\u2013Hadamard type for s\u2013convex functions with applications","volume":"15","author":"Khan","year":"2017","journal-title":"Open Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3418","DOI":"10.3934\/math.2022190","article-title":"Hermite-Hadamard like inequalities for fractional integral operator via convexity and quasi-convexity with their applications","volume":"7","author":"Nasir","year":"2022","journal-title":"AIMS Math."},{"key":"ref_17","first-page":"72","article-title":"New generalization of reverse Minkowski\u2019s inequality for fractional integral","volume":"1","author":"Aljaaidia","year":"2021","journal-title":"Adv. Theory. Nonlinear Anal. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"75","DOI":"10.30538\/psrp-easl2020.0053","article-title":"Reverse Hermite-Hadamard\u2019s inequalities using Q-fractional integral","volume":"3","author":"Aljaaidia","year":"2020","journal-title":"Eng. Appl. Sci. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Aljaaidi, T.A., Pachpatte, D.B., Shatanawi, W., Abdo, M.S., and Abodayeh, K. (2021). Generalized proportional fractional integral functional bounds in Minkowski\u2019s inequalities. Adv. Differ. Equ., 2021.","DOI":"10.1186\/s13662-021-03582-8"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1304","DOI":"10.3934\/math.2020089","article-title":"On n\u2013polynomial convexity and some related inequalities","volume":"5","author":"Toplu","year":"2020","journal-title":"AIMS Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1186\/1029-242X-2013-527","article-title":"Some new judgement theorems of Schur geometric and schur harmonic convexities for a class of symmetric function","volume":"2013","author":"Shi","year":"2013","journal-title":"J. Inequal. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1294","DOI":"10.1016\/j.jmaa.2007.02.016","article-title":"Generalized convexity and inequalities","volume":"335","author":"Anderson","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1186\/s13660-020-02393-x","article-title":"New Hermite\u2013Hadamard type inequalities for n\u2013polynomial harmonically convex functions","volume":"2020","author":"Awan","year":"2020","journal-title":"J. Inequal. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1811","DOI":"10.18576\/amis\/100522","article-title":"Harmonic variational inequalities","volume":"10","author":"Noor","year":"2016","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_25","first-page":"10","article-title":"Some implicit methods for solving harmonic variational inequalities","volume":"12","author":"Noor","year":"2016","journal-title":"Inter. J. Anal. Appl."},{"key":"ref_26","first-page":"527","article-title":"Some Hermite\u2013Hadamard\u2019s inequality functions whose exponentials are convex","volume":"60","author":"Dragomir","year":"2015","journal-title":"Babes Bolyai Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"405","DOI":"10.18576\/amis\/120215","article-title":"Hermite\u2013Hadamard type inequalities for exponentially convex functions","volume":"12","author":"Awan","year":"2018","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1186\/s13660-020-02349-1","article-title":"Exponential type convexity and some related inequalities","volume":"2020","author":"Kadakal","year":"2020","journal-title":"J. Inequal. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/j.camwa.2009.08.002","article-title":"Refinements of Hadamard-type inequalities for quasi-convex functions with applications to trapezoidal formula and to special means","volume":"59","author":"Alomari","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"705","DOI":"10.22436\/jnsa.009.02.32","article-title":"Several complementary inequalities to inequalities of Hermite\u2013Hadamard type for s\u2013convex functions","volume":"9","author":"Chen","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"14282","DOI":"10.3934\/math.2022787","article-title":"On inequalities of Hermite-Hadamard type via n-polynomial exponential type s-convex functions","volume":"7","author":"Samraiz","year":"2022","journal-title":"AIMS Math."},{"key":"ref_32","first-page":"103","article-title":"Some new Hermite\u2013Hadamard type inequalities and their applications","volume":"56","author":"Kashuri","year":"2019","journal-title":"Stud. Sci. Math. Hung."},{"key":"ref_33","first-page":"67","article-title":"New inequalities of Ostrowski\u2019s type for s-convex functions in the second sense with applications","volume":"27","author":"Set","year":"2012","journal-title":"Facta Univ. (Nis) Ser. Math."},{"key":"ref_34","first-page":"935","article-title":"Hermite\u2013Hadamard type inequalities for harmonically convex functions","volume":"43","year":"2014","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"13272","DOI":"10.3934\/math.2021768","article-title":"Some Ostrowski type inequalities via npolynomial exponentially s-convex functions and their applications","volume":"6","author":"Tariq","year":"2021","journal-title":"AIMS Math."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"6856","DOI":"10.3934\/math.2020440","article-title":"New inequalities via n-polynomial harmonically exponential type functions","volume":"5","author":"Geo","year":"2020","journal-title":"AIMS Math."},{"key":"ref_37","unstructured":"Mitrinovic, D.S., Pe\u0107aric, J.E., and Fink, A.M. (1993). Classical and New Inequalities in Analysis, Kluwer Academic."},{"key":"ref_38","first-page":"722","article-title":"Characterizations of classes of harmonic convex functions and applications","volume":"17","author":"Baloch","year":"2019","journal-title":"Int. J. Anal. Appl."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Abramowitz, M., and Stegun, I.A. (1965). Handbook of Mathematical Functions with Formulas. 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