{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T06:40:18Z","timestamp":1764571218845,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,7,25]],"date-time":"2025-07-25T00:00:00Z","timestamp":1753401600000},"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>A refined version of the q-Hermite\u2013Hadamard inequalities for strongly convex functions is introduced in this paper, utilizing both left and right q-integrals. Tighter bounds and more accurate estimates are derived by incorporating strong convexity. New q-trapezoidal and q-midpoint estimates are also presented to enhance the precision of the results. The improvements in the results compared to previous work are demonstrated through numerical examples in terms of precision and tighter bounds, and the advantages of using strongly convex functions are showcased.<\/jats:p>","DOI":"10.3390\/axioms14080576","type":"journal-article","created":{"date-parts":[[2025,7,25]],"date-time":"2025-07-25T12:56:43Z","timestamp":1753448203000},"page":"576","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["New Estimates of the q-Hermite\u2013Hadamard Inequalities via Strong Convexity"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5060-031X","authenticated-orcid":false,"given":"Chanokgan","family":"Sahatsathatsana","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5310-5127","authenticated-orcid":false,"given":"Pongsakorn","family":"Yotkaew","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,25]]},"reference":[{"key":"ref_1","first-page":"72","article-title":"Existence theorems and convergence of minimizing sequences in extremum problems with restictions","volume":"7","author":"Polyak","year":"1966","journal-title":"Sov. Math. Dokl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/s00010-010-0043-0","article-title":"Remarks on strongly convex functions","volume":"80","author":"Merentes","year":"2010","journal-title":"Aequat. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"83","DOI":"10.15352\/bjma\/1313362982","article-title":"Characterizations of inner product spaces be strongly convex functions","volume":"5","author":"Nikodem","year":"2011","journal-title":"Banach J. Math. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"85","DOI":"10.15352\/afa\/1399900197","article-title":"On strongly h-convex functions","volume":"2","author":"Angulo","year":"2011","journal-title":"Ann. Funct. Anal."},{"key":"ref_5","first-page":"123","article-title":"On Hermite-Hadamard type inequalities for strongly \u03c6-convex functions","volume":"39","author":"Sarikaya","year":"2015","journal-title":"Southeast Asian Bull. Math."},{"key":"ref_6","first-page":"1","article-title":"Sur deux limites ddune integrale definie","volume":"3","author":"Hermite","year":"1883","journal-title":"Mathesis"},{"key":"ref_7","first-page":"171","article-title":"\u00c9tude sur les propri\u00e9t\u00e9s des fonctions enti\u00e8res et en particulier d\u2019une fonction consid\u00e9r\u00e9e par riemann","volume":"9","author":"Hadamard","year":"1893","journal-title":"J. Math. Pures Appl."},{"key":"ref_8","unstructured":"Dragomir, S.S., and Pearce, C. (2003). Selected Topics on Hermite-Hadamard Inequalities and Applications, Elsevier."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.jksus.2016.09.007","article-title":"q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions","volume":"30","author":"Alp","year":"2018","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1007\/s10474-020-01025-6","article-title":"On q-Hermite\u2013Hadamard inequalities for general convex functions","volume":"162","author":"Bermudo","year":"2020","journal-title":"Acta Math. Hungar."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1515\/math-2021-0015","article-title":"On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions","volume":"19","author":"Ali","year":"2021","journal-title":"Open Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1186\/s13662-020-03163-1","article-title":"Quantum Hermite\u2013Hadamard-type inequalities for functions with convex absolute values of second qb-derivatives","volume":"2021","author":"Ali","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_13","first-page":"341","article-title":"Hermite\u2013Hadamard\u2019s type inequalities for co-ordinated convex functions on quantum integral","volume":"20","author":"Alp","year":"2020","journal-title":"Appl. Math. E-Notes"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1007\/s10957-020-01726-6","article-title":"Some new quantum Hermite\u2013Hadamard-like inequalities for coordinated convex functions","volume":"186","author":"Budak","year":"2020","journal-title":"J. Optim. Theory Appl."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Ding, Y., Kalsoom, H., and Wu, S. (2019). Some new quantum Hermite\u2013Hadamard-type estimates within a class of generalized (s, m)-preinvex functions. Symmetry, 11.","DOI":"10.3390\/sym11101283"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Jhanthanam, S., Tariboon, J., Ntouyas, S.K., and Nonlaopon, N. (2019). On q-Hermite\u2013Hadamard inequalities for differentiable convex functions. Mathematics, 7.","DOI":"10.3390\/math7070632"},{"key":"ref_17","first-page":"501","article-title":"Some quantum estimates of Hermite\u2013Hadamard inequalities for convex functions","volume":"7","author":"Liu","year":"2016","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_18","first-page":"675","article-title":"Some quantum estimates for Hermite\u2013Hadamard inequalities","volume":"251","author":"Noor","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1515\/ms-2023-0029","article-title":"A new version of q-Hermite\u2013Hadamard\u2019s midpoint and trapezoid type inequalities for convex functions","volume":"73","author":"Ali","year":"2023","journal-title":"Math. Slovaca"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1555","DOI":"10.18514\/MMN.2023.4200","article-title":"A new q-Hermite-Hadamard\u2019s inequality and estimates for midpoint type inequalities for convex functions","volume":"24","author":"Sitthiwirattham","year":"2023","journal-title":"Miskolc Math. Notes"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Chasreechai, S., Ali, M.A., Ashraf, M.A., Sitthiwirattham, T., Etemad, S., Sen, M.D.L., and Rezapour, S. (2023). On new estimates of q-Hermite\u2013Hadamard inequalities with applications in quantum calculus. Axioms, 12.","DOI":"10.3390\/axioms12010049"},{"key":"ref_22","first-page":"242","article-title":"Some quantum integral inequalities via preinvex functions","volume":"269","author":"Noor","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Ciurdariu, L., and Grecu, E. (2023). Several Quantum Hermite\u2013Hadamard-Type Integral Inequalities for Convex Functions. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7060463"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"5523","DOI":"10.3934\/math.2024268","article-title":"Some Hermite-Hadamard and midpoint type inequalities in symmetric quantum calculus","volume":"9","author":"Butt","year":"2024","journal-title":"AIMS Math."},{"key":"ref_25","first-page":"128345","article-title":"Hermite-Hadamard inequalities for quantum integrals: A unified approach","volume":"463","author":"Cardoso","year":"2024","journal-title":"Appl. Math. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"35","DOI":"10.22436\/jmcs.036.01.03","article-title":"On Hermite-Hadamard and Ostrowski type inequalities for strongly convex functions via quantum calculus with applications","volume":"36","author":"Sahatsathatsana","year":"2025","journal-title":"J. Math. Comput. SCI-JM"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"5661","DOI":"10.29020\/nybg.ejpam.v18i1.5661","article-title":"On the extension of q-Hermite-Hadamard inequalities for strong convexity","volume":"18","author":"Sahatsathatsana","year":"2025","journal-title":"Eur. J. Pure Appl. Math."},{"key":"ref_28","first-page":"193","article-title":"On a q-definite integrals","volume":"4","author":"Jackson","year":"1910","journal-title":"Q. J. Pure Appl. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"305","DOI":"10.2307\/2370183","article-title":"q-difference equations","volume":"32","author":"Jackson","year":"1910","journal-title":"Am. J. Math."},{"key":"ref_30","unstructured":"Kac, V.G., and Cheung, P. (2000). Quantum Calculus, Spinger."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Ernst, T. (2012). A Comprehensive Treatment of q-Calculus, Springer.","DOI":"10.1007\/978-3-0348-0431-8"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"487","DOI":"10.2991\/jnmp.2003.10.4.5","article-title":"A method for q-calculus","volume":"10","author":"Ernst","year":"2003","journal-title":"J. Nonlinear Math. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1186\/1687-1847-2013-282","article-title":"Quantum calculus on finite intervals and applications to impulsive difference equations","volume":"2013","author":"Tariboon","year":"2013","journal-title":"Adv. Differ. Equ."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1186\/1029-242X-2014-121","article-title":"Quantum integral inequalities on finite intervals","volume":"2014","author":"Tariboon","year":"2014","journal-title":"J. Inequal. Appl."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Anastassiou, G.A. (2011). Intelligent Mathematics: Computational Analysis, Springer.","DOI":"10.1007\/978-3-642-17098-0"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/576\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:16:05Z","timestamp":1760033765000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/576"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,25]]},"references-count":35,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["axioms14080576"],"URL":"https:\/\/doi.org\/10.3390\/axioms14080576","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,7,25]]}}}