{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:11:36Z","timestamp":1760145096985,"version":"build-2065373602"},"reference-count":59,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,6,21]],"date-time":"2024-06-21T00:00:00Z","timestamp":1718928000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University Researchers Supporting Project","award":["PNURSP2024R183"],"award-info":[{"award-number":["PNURSP2024R183"]}]},{"name":"Princess Nourah bint Abdulrahman University","award":["PNURSP2024R183"],"award-info":[{"award-number":["PNURSP2024R183"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Inequalities involving fractional operators have been an active area of research, which is crucial in establishing bounds, estimates, and stability conditions for solutions to fractional integrals. In this paper, we initially presented a new class that is known as coordinated left and right \u210f-pre-invex interval-valued mappings (C\u00b7L\u00b7R-\u210f-pre-invex \u0399\u00b7V-M), as well classical convex and nonconvex are also obtained. This newly defined class enabled us to derive novel inequalities, such as Hermite\u2013Hadamard and Pachpatte\u2019s type inequalities. Furthermore, the obtained results allowed us to recapture several special cases of known results for different parameter choices, which can be applications of the main results. Finally, we discussed the validity of the main outcomes.<\/jats:p>","DOI":"10.3390\/axioms13070417","type":"journal-article","created":{"date-parts":[[2024,6,21]],"date-time":"2024-06-21T05:33:28Z","timestamp":1718948008000},"page":"417","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Some New Estimations of Left and Right Interval Fractional Pachpatte\u2019s Type Integral Inequalities via Rectangle Plane"],"prefix":"10.3390","volume":"13","author":[{"given":"Azzh Saad","family":"Alshehry","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1234-7939","authenticated-orcid":false,"given":"Loredana","family":"Ciurdariu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Politehnica University of Timi\u0219oara, 300006 Timisoara, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4795-3469","authenticated-orcid":false,"given":"Yaser","family":"Saber","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science Al-Zulfi, Majmaah University, P.O. Box 66, Al-Majmaah 11952, Saudi Arabia"}]},{"given":"Amal F.","family":"Soliman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia"},{"name":"Department of Basic Science, Benha Faculty of Engineering, Benha University, Banha 13511, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,21]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Todinov, M. (2021). Interpretation of Algebraic Inequalities: Practical Engineering Optimisation and Generating New Knowledge, CRC Press.","key":"ref_1","DOI":"10.1201\/9781003199830"},{"doi-asserted-by":"crossref","unstructured":"Qin, Y. (2016). Integral and Discrete Inequalities and Their Applications, Birkh\u00e4user.","key":"ref_2","DOI":"10.1007\/978-3-319-33301-4"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"112692","DOI":"10.1016\/j.chaos.2022.112692","article-title":"Some new concepts related to fuzzy fractional calculus for up and down convex fuzzy-number valued functions and inequalities","volume":"164","author":"Khan","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"6423","DOI":"10.1109\/TAC.2023.3238849","article-title":"Non-fragile finite-time stabilization for discrete mean-field stochastic systems","volume":"68","author":"Zhang","year":"2023","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"140202","DOI":"10.1007\/s11432-023-3982-y","article-title":"Online Pareto optimal control of mean-field stochastic multi-player systems using policy iteration","volume":"67","author":"Jiang","year":"2024","journal-title":"Sci. China Inf. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"155417","DOI":"10.1103\/PhysRevB.109.155417","article-title":"Valley quantum interference modulated by hyperbolic shear polaritons","volume":"109","author":"Jia","year":"2023","journal-title":"Phys. Rev. B"},{"doi-asserted-by":"crossref","unstructured":"Ullah, N., Khan, M.B., Aloraini, N., and Trean\u021b\u01ce, S. (2023). Some New Estimates of Fixed Point Results under Multi-Valued Mappings in G-Metric Spaces with Application. Symmetry, 15.","key":"ref_7","DOI":"10.3390\/sym15020517"},{"doi-asserted-by":"crossref","unstructured":"Tian, F., Liu, Z., Zhou, J., Chen, L., and Feng, X.T. (2024). Quantifying Post-peak Behavior of Rocks with Type-I, Type-II, and Mixed Fractures by Developing a Quasi-State-Based Peridynamics. Rock Mech. Rock Eng., 1\u201337.","key":"ref_8","DOI":"10.1007\/s00603-024-03788-8"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2316107","DOI":"10.1080\/17538947.2024.2316107","article-title":"Mitigation of tropospheric delay induced errors in TS-InSAR ground deformation monitoring","volume":"17","author":"Guo","year":"2024","journal-title":"Int. J. Digit. Earth"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"129969","DOI":"10.1016\/j.jhydrol.2023.129969","article-title":"Study on optimization and combination strategy of multiple daily runoff prediction models coupled with physical mechanism and LSTM","volume":"624","author":"Guo","year":"2023","journal-title":"J. Hydrol."},{"doi-asserted-by":"crossref","unstructured":"Chang, X., Guo, J., Qin, H., Huang, J., Wang, X., and Ren, P. (2024). Single-Objective and Multi-Objective Flood Interval Forecasting Considering Interval Fitting Coefficients. Water Resour. Manag., 1\u201320.","key":"ref_11","DOI":"10.1007\/s11269-024-03848-2"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/j.spl.2019.03.006","article-title":"Jensen inequality for superlinear expectations","volume":"151","author":"Lin","year":"2019","journal-title":"Stat. Probab. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/BF00934845","article-title":"Application of Jensen\u2019s inequality to adaptive suboptimal design","volume":"32","author":"White","year":"1980","journal-title":"J. Optim. Theory Appl."},{"doi-asserted-by":"crossref","unstructured":"Dou, J., Liu, J., Wang, Y., Zhi, L., Shen, J., and Wang, G. (2023). Surface activity, wetting, and aggregation of a perfluoropolyether quaternary ammonium salt surfactant with a hydroxyethyl group. Molecules, 28.","key":"ref_14","DOI":"10.3390\/molecules28207151"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1007\/s11235-023-01059-5","article-title":"LMCA: A lightweight anomaly network traffic detection model integrating adjusted mobilenet and coordinate attention mechanism for IoT","volume":"84","author":"Han","year":"2023","journal-title":"Telecommun. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2206264","DOI":"10.1002\/advs.202206264","article-title":"Electric-Field-Driven Printed 3D Highly Ordered Microstructure with Cell Feature Size Promotes the Maturation of Engineered Cardiac Tissues","volume":"10","author":"Zhang","year":"2023","journal-title":"Adv. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1587\/transinf.2022DLP0073","article-title":"An improved BPNN method based on probability density for indoor location","volume":"106","author":"Fei","year":"2023","journal-title":"IEICE Trans. Inf. Syst."},{"unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier.","key":"ref_18"},{"doi-asserted-by":"crossref","unstructured":"Atangana, A., and Baleanu, D. (2016). New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. arXiv.","key":"ref_19","DOI":"10.2298\/TSCI160111018A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"7411","DOI":"10.1109\/TPEL.2021.3139646","article-title":"Single-Stage Multi-Input Buck Type High-Frequency Link\u2019s Inverters with Series and Simultaneous Power Supply","volume":"37","author":"Chen","year":"2021","journal-title":"IEEE Trans. Power Electron."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"12763","DOI":"10.1109\/TPEL.2022.3176377","article-title":"Single-stage multi-input buck type high-frequency link\u2019s inverters with multiwinding and time-sharing power supply","volume":"37","author":"Chen","year":"2022","journal-title":"IEEE Trans. Power Electron."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2996","DOI":"10.1007\/s11431-022-2394-4","article-title":"SVM strategy and analysis of a three-phase quasi-Z-source inverter with high voltage transmission ratio","volume":"66","author":"Chen","year":"2023","journal-title":"Sci. China Technol. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"11397","DOI":"10.1016\/j.jfranklin.2023.08.037","article-title":"A robust observer based on the nonlinear descriptor systems application to estimate the state of charge of lithium-ion batteries","volume":"360","author":"Meng","year":"2023","journal-title":"J. Frankl. Inst."},{"doi-asserted-by":"crossref","unstructured":"Kashuri, A., Almalki, Y., Mahnashi, A.M., and Sahoo, S.K. (2023). New estimates on Hermite\u2013Hadamard type inequalities via generalized tempered fractional integrals for convex functions with applications. Fractal Fract., 7.","key":"ref_24","DOI":"10.3390\/fractalfract7080579"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"110706","DOI":"10.1016\/j.knosys.2023.110706","article-title":"CLVIN: Complete language-vision interaction network for visual question answering","volume":"275","author":"Chen","year":"2023","journal-title":"Knowl.-Based Syst."},{"doi-asserted-by":"crossref","unstructured":"Junjua, M.U.D., Qayyum, A., Munir, A., Budak, H., Saleem, M.M., and Supadi, S.S. (2024). A study of some new Hermite Hadamard inequalities via specific convex functions with applications. Mathematics, 12.","key":"ref_26","DOI":"10.3390\/math12030478"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2227780","DOI":"10.1080\/09540091.2023.2227780","article-title":"A multimodal hybrid parallel network intrusion detection model","volume":"35","author":"Shi","year":"2023","journal-title":"Connect. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2153","DOI":"10.2298\/FIL1806153U","article-title":"On generalization of trapezoid type inequalities for s-convex functions with generalized fractional integral operators","volume":"32","author":"Usta","year":"2018","journal-title":"Filomat"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"110084","DOI":"10.1016\/j.patcog.2023.110084","article-title":"MPCCT: Multimodal vision-language learning paradigm with context-based compact Transformer","volume":"147","author":"Chen","year":"2024","journal-title":"Pattern Recognit."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"19391","DOI":"10.3934\/math.2023989","article-title":"New inequalities via Caputo-Fabrizio integral operator with applications","volume":"8","author":"Yang","year":"2023","journal-title":"Aims Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/09540091.2023.2257399","article-title":"NAS-YOLOX: A SAR ship detection using neural architecture search and multi-scale attention","volume":"35","author":"Wang","year":"2023","journal-title":"Connect. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2627","DOI":"10.1007\/s10773-021-04739-7","article-title":"Trapezoidal (p,q)-integral inequalities related to (\u03b71,\u03b72)-convex functions with applications","volume":"60","author":"Klasoom","year":"2021","journal-title":"Int. J. Theor. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-020-03121-x","article-title":"Generalized trapezium-type inequalities in the settings of fractal sets for functions having generalized convexity property","volume":"2020","author":"Khan","year":"2020","journal-title":"Adv. Differ. Equ."},{"doi-asserted-by":"crossref","unstructured":"Chuai, Y., Dai, B., Liu, X., Hu, M., Wang, Y., and Zhang, H. (2023). Association of vitamin K, fibre intake and progression of periodontal attachment loss in American adults. BMC Oral Health, 23.","key":"ref_34","DOI":"10.1186\/s12903-023-02929-9"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"570","DOI":"10.1186\/s13662-020-03028-7","article-title":"On the Hermite-Hadamard inequalities for interval-valued coordinated convex functions","volume":"2020","author":"Zhao","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"4727","DOI":"10.1007\/s00784-023-05100-4","article-title":"Associations between vitamin D levels and periodontal attachment loss","volume":"27","author":"Liu","year":"2023","journal-title":"Clin. Oral Investig."},{"doi-asserted-by":"crossref","unstructured":"Lai, K.K., Mishra, S.K., Bisht, J., and Hassan, M. (2022). Hermite-Hadamard type inclusions for interval-valued coordinated preinvex functions. Symmetry, 14.","key":"ref_37","DOI":"10.3390\/sym14040771"},{"doi-asserted-by":"crossref","unstructured":"Wannalookkhee, F., Nonlaopon, K., Tariboon, J., and Ntouyas, S.K. (2021). On Hermite-Hadamard type inequalities for coordinated convex functions via (p,q)-calculus. Mathematics, 9.","key":"ref_38","DOI":"10.22541\/au.163332914.42587950\/v1"},{"doi-asserted-by":"crossref","unstructured":"Kalsoom, H., Rashid, S., Idrees, M., Safdar, F., Akram, S., Baleanu, D., and Chu, Y.-M. (2020). Post quantum integral inequalities of Hermite-Hadamard-type associated with coordinated higher order generalized strongly pre-invex and quasi-pre-invex mappings. Symmetry, 12.","key":"ref_39","DOI":"10.3390\/sym12030443"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1016\/j.jksus.2016.06.003","article-title":"On the Hadamard\u2019s type inequalities for coordinated convex functions via fractional integrals","volume":"29","author":"Akkurt","year":"2017","journal-title":"J. King Saud. Univ. Sci."},{"doi-asserted-by":"crossref","unstructured":"Shi, F., Ye, G., Zhao, D., and Liu, W. (2020). Some fractional Hermite-Hadamard type inequalities for interval valued functions. Mathematics, 8.","key":"ref_41","DOI":"10.3390\/math8040534"},{"doi-asserted-by":"crossref","unstructured":"Saeed, T., C\u02d8atas, A., Khan, M.B., and Alshehri, A.M. (2023). Some new fractional inequalities for coordinated convexity over convex set pertaining to fuzzy-number-valued settings governed by fractional integrals. Fractal Fract., 7.","key":"ref_42","DOI":"10.20944\/preprints202310.1591.v1"},{"doi-asserted-by":"crossref","unstructured":"Wu, X., Wang, J., and Zhang, J. (2019). Hermite-Hadamard-type inequalities for convex functions via the fractional integrals with exponential kernel. Mathematics, 7.","key":"ref_43","DOI":"10.3390\/math7090845"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.cam.2018.12.030","article-title":"Hermite-Hadamard, Hermite Hadamard-Fej\u2019er, Dragomir-Agarwal and Pachpatte type inequalities for convex functions via new fractional integrals","volume":"353","author":"Ahmad","year":"2019","journal-title":"J. Comput. Appl. Math."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"113274","DOI":"10.1016\/j.chaos.2023.113274","article-title":"On fuzzy fractional integral operators having exponential kernels and related certain inequalities for exponential trigonometric convex fuzzy-number valued mappings","volume":"169","author":"Khan","year":"2023","journal-title":"Chaos Soliton Fract."},{"doi-asserted-by":"crossref","unstructured":"Khan, M.B., Nwaeze, E.R., Lee, C.-C., Zaini, H.G., Lou, D.-C., and Hakami, K.H. (2023). Weighted Fractional Hermite\u2013Hadamard Integral Inequalities for up and down \u0512-Convex Fuzzy Mappings over Coordinates. Mathematics, 11.","key":"ref_46","DOI":"10.3390\/math11244974"},{"doi-asserted-by":"crossref","unstructured":"Khan, M.B., Mac\u00edas-D\u00edaz, J.E., Althobaiti, A., and Althobaiti, S. (2023). Some New Properties of Exponential Trigonometric Convex Functions Using Up and Down Relations over Fuzzy Numbers and Related Inequalities through Fuzzy Fractional Integral Operators Having Exponential Kernels. Fractal Fract., 7.","key":"ref_47","DOI":"10.3390\/fractalfract7070567"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"114196","DOI":"10.1016\/j.chaos.2023.114196","article-title":"Riemann Liouville fractional-like integral operators, convex-like real-valued mappings and their applications over fuzzy domain","volume":"177","author":"Khan","year":"2023","journal-title":"Chaos Solitons Fractals"},{"doi-asserted-by":"crossref","unstructured":"Alreshidi, N.A., Khan, M.B., Breaz, D., and Cotirla, L.-I. (2023). New Versions of Fuzzy-Valued Integral Inclusion Over p-Convex Fuzzy Number-Valued Mappings and Related Fuzzy Aumman\u2019s Integral Inequalities. Symmetry, 15.","key":"ref_49","DOI":"10.3390\/sym15122123"},{"doi-asserted-by":"crossref","unstructured":"Saeed, T., Nwaeze, E.R., Khan, M.B., and Hakami, K.H. (2024). New version of fractional Pachpatte-type integral inequalities via coordinated h-convexity via left and right order relation. Fractal Fract., 8.","key":"ref_50","DOI":"10.3390\/fractalfract8030125"},{"doi-asserted-by":"crossref","unstructured":"Stojiljkovic, V., Ramaswamy, R., Abdelnaby, O.A.A., and Radenovi\u0107, S. (2022). Some novel inequalities for LR-(k,h-m)-p convex interval valued functions by means of pseudo order relation. Fractal Fract., 6.","key":"ref_51","DOI":"10.3390\/fractalfract6120726"},{"key":"ref_52","first-page":"18","article-title":"Hermite\u2013Hadamard\u2013type fractional\u2013integral inequalities for (p, h)\u2013convex fuzzy\u2013interval\u2013valued mappings","volume":"5","author":"Stojiljkovic","year":"2023","journal-title":"Electron. J. Math"},{"doi-asserted-by":"crossref","unstructured":"Khan, M.B., C\u0103ta\u015f, A., and Alsalami, O.M. (2022). Some New Estimates on Coordinates of Generalized Convex Interval-Valued Functions. Fractal Fract., 6.","key":"ref_53","DOI":"10.3390\/fractalfract6080415"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1186\/1029-242X-2013-227","article-title":"On some Hadamard-type inequalities for (\u210e1,\u210e2)-preinvex functions on the co-ordinates","volume":"2013","year":"2013","journal-title":"J. Inequal. Appl."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/j.fss.2014.04.005","article-title":"Fractional calculus for interval-valued functions","volume":"265","author":"Lupulescu","year":"2015","journal-title":"Fuzzy Set. Syst."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"1081","DOI":"10.1515\/math-2021-0067","article-title":"Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions","volume":"19","author":"Budak","year":"2021","journal-title":"Open Math."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.fss.2019.10.006","article-title":"Chebyshev type inequalities for interval-valued functions","volume":"396","author":"Zhao","year":"2019","journal-title":"Fuzzy Sets Syst."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"15659","DOI":"10.3934\/math.2022857","article-title":"Some new Riemann-Liouville fractional integral inequalities for interval-valued mappings","volume":"7","author":"Khan","year":"2022","journal-title":"AIMS Math"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/j.fss.2020.06.003","article-title":"Jensen\u2019s inequalities for set-valued and fuzzy set-valued functions","volume":"404","author":"Zhang","year":"2020","journal-title":"Fuzzy Sets Syst."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/417\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:02:14Z","timestamp":1760108534000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/417"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,21]]},"references-count":59,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2024,7]]}},"alternative-id":["axioms13070417"],"URL":"https:\/\/doi.org\/10.3390\/axioms13070417","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,6,21]]}}}