{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,10]],"date-time":"2025-12-10T12:31:10Z","timestamp":1765369870363,"version":"build-2065373602"},"reference-count":68,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,9,29]],"date-time":"2021-09-29T00:00:00Z","timestamp":1632873600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>It is a familiar fact that convex and non-convex fuzzy mappings play a critical role in the study of fuzzy optimization. Due to the behavior of its definition, the idea of convexity plays a significant role in the subject of inequalities. The concepts of convexity and symmetry have a tight connection. We may use whatever we learn from one to the other, thanks to the significant correlation that has developed between both in recent years. Our aim is to consider a new class of fuzzy mappings (FMs) known as strongly preinvex fuzzy mappings (strongly preinvex-FMs) on the invex set. These FMs are more general than convex fuzzy mappings (convex-FMs) and preinvex fuzzy mappings (preinvex-FMs), and when generalized differentiable (briefly, G-differentiable), strongly preinvex-FMs are strongly invex fuzzy mappings (strongly invex-FMs). Some new relationships among various concepts of strongly preinvex-FMs are established and verified with the support of some useful examples. We have also shown that optimality conditions of G-differentiable strongly preinvex-FMs and the fuzzy functional, which is the sum of G-differentiable preinvex-FMs and non G-differentiable strongly preinvex-FMs, can be distinguished by strongly fuzzy variational-like inequalities and strongly fuzzy mixed variational-like inequalities, respectively. In the end, we have established and verified a strong relationship between the Hermite\u2013Hadamard inequality and strongly preinvex-FM. Several exceptional cases are also discussed. These inequalities are a very interesting outcome of our main results and appear to be new ones. The results in this research can be seen as refinements and improvements to previously published findings.<\/jats:p>","DOI":"10.3390\/sym13101816","type":"journal-article","created":{"date-parts":[[2021,10,11]],"date-time":"2021-10-11T01:59:47Z","timestamp":1633917587000},"page":"1816","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Fuzzy Mixed Variational-like and Integral Inequalities for Strongly Preinvex Fuzzy Mappings"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7450-8067","authenticated-orcid":false,"given":"Muhammad Bilal","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari Mohan","family":"Srivastava","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"},{"name":"Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku AZ1007, Azerbaijan"},{"name":"Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6837-8075","authenticated-orcid":false,"given":"Pshtiwan Othman","family":"Mohammed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2788-809X","authenticated-orcid":false,"given":"Juan L. G.","family":"Guirao","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Statistics, Technical University of Cartagena, Hospital de Marina, 30203 Cartagena, Spain"},{"name":"Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,29]]},"reference":[{"key":"ref_1","first-page":"2","article-title":"Existence theorems and convergence of minimizing sequences in extremum problems with restrictions","volume":"7","author":"Polyak","year":"1966","journal-title":"Sov. Math. Dokl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/BF00930577","article-title":"The nonlinear complementarity problem with applications, Part 2","volume":"4","author":"Karamardian","year":"1969","journal-title":"J. Optim. Theory Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1109\/LCSYS.2018.2851375","article-title":"On the exponentially stability of primal-dual gradeint dynamics","volume":"3","author":"Qu","year":"2019","journal-title":"IEEE Control Syst. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"83","DOI":"10.15352\/bjma\/1313362982","article-title":"Characterizations of inner product spaces by strongly convex functions","volume":"1","author":"Nikodem","year":"2011","journal-title":"Banach J. Math. Anal."},{"key":"ref_5","first-page":"1287","article-title":"On a problem connected with strongly convex functions","volume":"19","author":"Adamek","year":"2016","journal-title":"Math. Inequal. Appl."},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"405","DOI":"10.18576\/amis\/120215","article-title":"Hermite-Hadamard inequalities for exponentially convex functions","volume":"12","author":"Awan","year":"2018","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_8","first-page":"145","article-title":"On strongly (p, h)-convex functions","volume":"10","author":"Awan","year":"2019","journal-title":"TWMS J. Pure Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"15","DOI":"10.7494\/OpMath.2011.31.1.15","article-title":"On strongly midconvex functions","volume":"31","author":"Azcar","year":"2011","journal-title":"Opusc. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"584","DOI":"10.1007\/BF02309176","article-title":"A note on strongly convex and strongly quasi convex functions","volume":"60","author":"Jovanovic","year":"1966","journal-title":"Math. Notes"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1016\/0022-247X(81)90123-2","article-title":"On sufficiency of the Kuhn-Tucker conditions","volume":"80","author":"Hanson","year":"1980","journal-title":"J. Math. Anal. Appl."},{"key":"ref_12","first-page":"1","article-title":"What is invexity?","volume":"28","author":"Mond","year":"1986","journal-title":"Anziam J."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"901","DOI":"10.1006\/jmaa.1995.1057","article-title":"On invex sets and preinvex functions","volume":"189","author":"Mohan","year":"1995","journal-title":"J. Math. Anal. Appl."},{"key":"ref_14","first-page":"1","article-title":"On strongly generalized preinvex functions","volume":"6","author":"Noor","year":"2005","journal-title":"J. Inequal. Pure Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1016\/j.jmaa.2005.05.014","article-title":"Some characterization of strongly preinvex functions","volume":"316","author":"Noor","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_16","first-page":"12736","article-title":"Generalized preinvex functions and their properties","volume":"2006","author":"Noor","year":"2006","journal-title":"Int. J. Stoch. Anal."},{"key":"ref_17","first-page":"1","article-title":"Some Integral Inequalities for Generalized Convex Fuzzy-Interval-Valued Functions via Fuzzy Riemann Integrals","volume":"14","author":"Khan","year":"2021","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_18","first-page":"3","article-title":"Integral inequaliies for differentiable harmonic preinvex functions (survey)","volume":"7","author":"Noor","year":"2016","journal-title":"J. Inequal. Pure Appl. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/0022-247X(88)90113-8","article-title":"Preinvex functions in multiobjective optimization","volume":"136","author":"Weir","year":"1986","journal-title":"J. Math. Anal. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/0022-247X(85)90205-7","article-title":"Some properties of convex fuzzy sets","volume":"111","author":"Liu","year":"1985","journal-title":"J. Math. Anal. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/0165-0114(80)90025-1","article-title":"Convex fuzzy sets","volume":"3","author":"Lowen","year":"1980","journal-title":"Fuzzy Sets Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/0165-0114(92)90319-Y","article-title":"On fuzzy convexity and parametric fuzzy optimization","volume":"49","author":"Ammar","year":"1992","journal-title":"Fuzzy Sets Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1016\/S0165-0114(97)00273-X","article-title":"Some properties of convex fuzzy sets and convex fuzzy cones","volume":"106","author":"Ammar","year":"1999","journal-title":"Fuzzy Sets Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1080\/00207727808941724","article-title":"Operations on fuzzy numbers","volume":"9","author":"Dubois","year":"1978","journal-title":"Int. J. Syst. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0165-0114(83)80107-9","article-title":"Topological properties of fuzzy numbers","volume":"10","author":"Goetschel","year":"1983","journal-title":"Fuzzy Sets Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/0165-0114(92)90256-4","article-title":"Convex fuzzy mappings","volume":"48","author":"Nanda","year":"1992","journal-title":"Fuzzy Sets Syst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/S0165-0114(97)00210-8","article-title":"On convex and concave fuzzy mappings","volume":"103","author":"Syau","year":"1999","journal-title":"Fuzzy Sets Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/S0165-0114(96)00192-3","article-title":"Convexity and local Lipschitz continuity of fuzzy-valued mappings","volume":"93","author":"Furukawa","year":"1998","journal-title":"Fuzzy Sets Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/S0165-0114(01)00157-9","article-title":"A class of convex fuzzy mappings","volume":"129","author":"Yan","year":"2002","journal-title":"Fuzzy Sets Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/0165-0114(86)90026-6","article-title":"Elementary fuzzy calculus","volume":"18","author":"Goetschel","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/0165-0114(94)90011-6","article-title":"Fuzzy preinvex functions","volume":"64","author":"Noor","year":"1994","journal-title":"Fuzzy Sets Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1016\/S0165-0114(02)00527-4","article-title":"(\u03a61, \u03a62)-convex fuzzy mappings","volume":"138","author":"Syau","year":"2003","journal-title":"Fuzzy Sets Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1741","DOI":"10.1016\/j.camwa.2006.02.005","article-title":"Fuzzy Weirstrass theorem and convex fuzzy mappings","volume":"51","author":"Syau","year":"2006","journal-title":"Comput. Math. Appl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1016\/j.fss.2004.08.001","article-title":"Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations","volume":"151","author":"Bede","year":"2005","journal-title":"Fuzzy Sets Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1016\/S0377-2217(01)00393-9","article-title":"Generalized invex monotonicity","volume":"144","year":"2003","journal-title":"Eur. J. Oper. Res."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1311","DOI":"10.1016\/j.na.2008.12.005","article-title":"Generalized Hukuhara differentiability of interval-valued functions and interval differential equations","volume":"71","author":"Stefanini","year":"2009","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/S0898-1221(99)00044-9","article-title":"Preinvex fuzzy mappings","volume":"37","author":"Syau","year":"1999","journal-title":"Comp. Math. Appl."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1080\/02331939408843995","article-title":"Variational-like inequalities","volume":"30","author":"Noor","year":"1994","journal-title":"Optimization"},{"key":"ref_40","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_41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2020\/8262860","article-title":"On new modifications governed by quantum Hahn\u2019s integral operator pertaining to fractional calculus","volume":"2020","author":"Rashid","year":"2020","journal-title":"J. Funct. Spaces"},{"key":"ref_42","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":"7","author":"Hadamard","year":"1893","journal-title":"J. Math. Pures Appl."},{"key":"ref_43","first-page":"82","article-title":"Sur deux limites d\u2019une int\u00e9grale d\u00e9finie","volume":"3","author":"Hermite","year":"1883","journal-title":"Mathesis"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/2251-7456-7-22","article-title":"A new generalization of some integral inequalities for (\u03b1, m)-convex functions","volume":"7","author":"Iscan","year":"2013","journal-title":"Math. Sci."},{"key":"ref_45","first-page":"935","article-title":"Hermite\u2013Hadamard type inequalities for harmonically convex functions","volume":"43","author":"Iscan","year":"2014","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_46","first-page":"137","article-title":"Hermite\u2013Hadamard type inequalities for p-convex functions","volume":"11","author":"Iscan","year":"2016","journal-title":"Int. J. Anal. Appl."},{"key":"ref_47","first-page":"1","article-title":"On some inequalities for convex functions","volume":"6","author":"Pachpatte","year":"2003","journal-title":"RGMIA Res. Rep. Coll."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/0165-0114(87)90029-7","article-title":"Fuzzy differential equations","volume":"24","author":"Kaleva","year":"1987","journal-title":"Fuzzy Sets Syst."},{"key":"ref_49","unstructured":"Kulish, U., and Miranker, W. (2014). Computer Arithmetic in Theory and Practice, Academic Press."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Osuna-G\u2019omez, R., Jim\u00b4enez-Gamero, M.D., Chalco-Cano, Y., and Rojas-Medar, M.A. (2004). Hadamard and Jensen Inequalities for s\u2212Convex Fuzzy Processes. Soft Methodology and Random Information Systems (Advances in Soft Computing), Springer.","DOI":"10.1007\/978-3-540-44465-7_80"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.fss.2017.02.001","article-title":"Jensen\u2019s inequality type integral for fuzzy-interval-valued functions","volume":"327","author":"Costa","year":"2017","journal-title":"Fuzzy Sets Syst."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.fss.2018.04.012","article-title":"Opial-type inequalities for interval-valued functions","volume":"358","author":"Costa","year":"2019","journal-title":"Fuzzy Sets Syst."},{"key":"ref_53","unstructured":"Moore, R.E. (1966). Interval Analysis, Prentice Hall."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.ins.2017.08.055","article-title":"Some integral inequalities for fuzzy-interval-valued functions","volume":"420","author":"Costa","year":"2017","journal-title":"Inf. Sci."},{"key":"ref_55","first-page":"6","article-title":"New Hermite-Hadamard Type Inequalities for (h1, h2)-Convex Fuzzy-Interval-Valued Functions","volume":"2021","author":"Khan","year":"2021","journal-title":"Adv. Differ. Equat."},{"key":"ref_56","first-page":"1","article-title":"New Hermite\u2013Hadamard and Jensen inequalities for log-s-convex fuzzy-interval-valued functions in the second sense","volume":"2021","author":"Liu","year":"2021","journal-title":"Complex Intell. Syst."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/s44196-021-00004-1","article-title":"New Hermite-Hadamard and Jensen Inequalities for Log-h-Convex Fuzzy-Interval-Valued Functions","volume":"14","author":"Khan","year":"2021","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1403","DOI":"10.2991\/ijcis.d.210409.001","article-title":"Some New Classes of Preinvex Fuzzy-Interval-Valued Functions and Inequalities","volume":"14","author":"Khan","year":"2021","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Khan, M.B., Mohammed, P.O., Noor, M.A., and Hamed, Y.S. (2021). New Hermite\u2013Hadamard inequalities in fuzzy-interval fractional calculus and related inequalities. Symmetry, 13.","DOI":"10.3390\/sym13040673"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"6552","DOI":"10.3934\/mbe.2021325","article-title":"Fuzzy Integral Inequalities on Coordinates of Convex Fuzzy Interval-Valued Functions","volume":"18","author":"Khan","year":"2021","journal-title":"Math. Biosci. Eng."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"10964","DOI":"10.3934\/math.2021637","article-title":"New Fuzzy-Interval Inequalities in Fuzzy-Interval Fractional Calculus by Means of Fuzzy Order Relation","volume":"6","author":"Khan","year":"2021","journal-title":"AIMS Math."},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"Khan, M.B., Mohammed, P.O., Noor, M.A., Baleanu, D., and Guirao, J.L.G. (2021). Some New Fractional Estimates of Inequalities for LR-p-Convex Interval-Valued Functions by Means of Pseudo Order Relation. Axioms, 10.","DOI":"10.3390\/axioms10030175"},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., and El-Deeb, S.M. (2021). Fuzzy differential subordinations based upon the Mittag-Leffler type Borel distribution. Symmetry, 13.","DOI":"10.3390\/sym13061023"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"35","DOI":"10.2478\/amns.2020.2.00012","article-title":"A modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives","volume":"5","author":"Touchent","year":"2020","journal-title":"Appl. Math. Nonlinear Sci."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"49","DOI":"10.2478\/amns.2020.2.00013","article-title":"Normal complex contact metric manifolds admitting a semi symmetric metric connection","volume":"5","author":"Vanli","year":"2020","journal-title":"Appl. Math. Nonlinear Sci."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"67","DOI":"10.2478\/amns.2020.2.00014","article-title":"Vortex Theory for Two Dimensional Boussinesq Equations","volume":"5","author":"Sharifi","year":"2020","journal-title":"Appl. Math. Nonlinear Sci."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"85","DOI":"10.2478\/amns.2020.2.00017","article-title":"On Solutions of Fractional order Telegraph Partial Differential Equation by Crank-Nicholson Finite Difference Method","volume":"5","author":"Nandappa","year":"2020","journal-title":"Appl. Math. Nonlinear Sci."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"99","DOI":"10.2478\/amns.2020.2.00018","article-title":"Degree Sequence of Graph Operator for some Standard Graphs","volume":"5","author":"Harisha","year":"2020","journal-title":"Appl. Math. Nonlinear Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/10\/1816\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:07:06Z","timestamp":1760166426000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/10\/1816"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,29]]},"references-count":68,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2021,10]]}},"alternative-id":["sym13101816"],"URL":"https:\/\/doi.org\/10.3390\/sym13101816","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,9,29]]}}}