{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T15:47:09Z","timestamp":1772293629135,"version":"3.50.1"},"reference-count":45,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,2]],"date-time":"2020-02-02T00:00:00Z","timestamp":1580601600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61673169"],"award-info":[{"award-number":["61673169"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In Hilbert space, we develop a novel framework to study for two new classes of convex function depending on arbitrary non-negative function, which is called a predominating \u210f-convex function and predominating quasiconvex function, with respect to    \u03b7   , are presented. To ensure the symmetry of data segmentation and with the discussion of special cases, it is shown that these classes capture other classes of    \u03b7   -convex functions,    \u03b7   -quasiconvex functions, strongly \u210f-convex functions of higher-order and strongly quasiconvex functions of a higher order, etc. Meanwhile, an auxiliary result is proved in the sense of    \u03ba   -fractional integral operator to generate novel variants related to the Hermite\u2013Hadamard type for     p t h    -order differentiability. It is hoped that this research study will open new doors for in-depth investigation in convexity theory frameworks of a varying nature.<\/jats:p>","DOI":"10.3390\/sym12020222","type":"journal-article","created":{"date-parts":[[2020,2,3]],"date-time":"2020-02-03T01:25:51Z","timestamp":1580693151000},"page":"222","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":45,"title":["New Multi-Parametrized Estimates Having pth-Order Differentiability in Fractional Calculus for Predominating \u210f-Convex Functions in Hilbert Space"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7137-1720","authenticated-orcid":false,"given":"Saima","family":"Rashid","sequence":"first","affiliation":[{"name":"Department of Mathematics, Government College University, Faisalabad 38000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5835-3349","authenticated-orcid":false,"given":"Humaira","family":"Kalsoom","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China"}]},{"given":"Zakia","family":"Hammouch","sequence":"additional","affiliation":[{"name":"Faculty of Science and Techniques Moulay Ismail University of Meknes, 52000 Errachidia, Morocco"}]},{"given":"Rehana","family":"Ashraf","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Lahore College for Women University, Jhangh Campus, Lahore 54500, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0944-2134","authenticated-orcid":false,"given":"Yu-Ming","family":"Chu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Huzhou University, Huzhou 313000, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2000). Applications of Fractional Calculus in Physics, Word Scientific.","DOI":"10.1142\/9789812817747"},{"key":"ref_2","unstructured":"Kilbas, A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_3","unstructured":"K\u00d6se, K. (2012). Signal and Image Processing Algorithims Using Interval Convex Programming and and Sparsity. [Ph.D. Thesis, Engineering and Science of Bilkent University]."},{"key":"ref_4","unstructured":"Miller, S., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley."},{"key":"ref_5","unstructured":"Podlubni, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1140","DOI":"10.1016\/j.cnsns.2010.05.027","article-title":"Recent history of fractional calculus","volume":"16","author":"Machado","year":"2011","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/S0045-7825(98)00108-X","article-title":"Approximate analytical solution for seepage flow withfractional derivatives in porous media","volume":"167","author":"He","year":"1998","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"699","DOI":"10.1016\/S0020-7462(98)00048-1","article-title":"Variational iteration method-a kind of non-linearanalytical technique: Some examples","volume":"34","author":"He","year":"1999","journal-title":"Int. J. Nonl. Mech."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Nie, D., Rashid, S., Akdemir, A.O., Baleanu, D., and Liu, J.-B. (2019). On some new weighted inequalities for differentiable exponentially convex and exponentially quasi-convex functions with applications. Mathematics, 7.","DOI":"10.3390\/math7080727"},{"key":"ref_10","first-page":"26","article-title":"An inequality related to \u03b7-convex functions (II)","volume":"6","author":"Gordji","year":"2015","journal-title":"Int. J. Nonlinear Anal. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Rashid, S., Abdeljawad, T., Jarad, F., and Noor, M.A. (2019). Some estimates for generalized Riemann-Liouville fractional integrals of exponentially convex functions and their applications. Mathematics, 7.","DOI":"10.3390\/math7090807"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Rashid, S., Noor, M.A., and Noor, K.I. (2019). Inequalities pertaining fractional approach through exponentially convex functions. Fractal Fract., 3.","DOI":"10.3390\/fractalfract3030037"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Rashid, S., Noor, M.A., Noor, K.I., and Akdemir, A.O. (2019). Some new generalizations for exponentially s-convex functions and inequalities via fractional operators. Fractal Fract., 3.","DOI":"10.3390\/fractalfract3020024"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2403","DOI":"10.1016\/j.mcm.2011.12.048","article-title":"Hermite\u2013Hadamard\u2019s inequalities for fractional integrals and related fractional inequalities","volume":"57","author":"Sarikaya","year":"2013","journal-title":"Math. Comput. Model."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Rashid, S., Noor, M.A., and Noor, K.I. (2019). New Estimates for Exponentially Convex Functions via Conformable Fractional Operator. Fractal Fract., 3.","DOI":"10.3390\/fractalfract3020019"},{"key":"ref_16","first-page":"1","article-title":"Some generalize Riemann-Liouville fractional estimates involving functions having exponentially convexity property","volume":"51","author":"Rashid","year":"2019","journal-title":"Punjab. Univ. J. Math."},{"key":"ref_17","first-page":"464","article-title":"Fractional exponentially m-convex functions and inequalities","volume":"17","author":"Rashid","year":"2019","journal-title":"Int. J. Anal. Appl."},{"key":"ref_18","first-page":"72","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_19","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/BF00930577","article-title":"The nonlinear complementarity problems with applications, Part 2","volume":"4","author":"Karamardian","year":"1969","journal-title":"J. Optim. Theory Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"714","DOI":"10.1137\/S1052623494250415","article-title":"Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities","volume":"6","author":"Zu","year":"1996","journal-title":"SIAM J. Optim."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"83","DOI":"10.15352\/bjma\/1313362982","article-title":"Characterizations of inner product spaces by strongly convex functions","volume":"5","author":"Nikodem","year":"2011","journal-title":"Banach J. Math. Anal."},{"key":"ref_22","unstructured":"Bassily, R., Belkin, M., and Ma, S. (2018). On exponential convergence of SGD in non-convex over-parametrized learning. arXiv."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Rashid, S., Latif, M.A., Hammouch, Z., and Chu, Y.-M. (2019). Fractional integral inequalities for strongly h-preinvex functions for a kth order differentiable functions. Symmetry, 11.","DOI":"10.3390\/sym11121448"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Kalsoom, H., Rashid, S., Idrees, M., Chu, Y.-M., and Baleanu, D. (2020). Two-variable quantum integral inequalities of Simpson-type based on higher-order generalized strongly preinvex and quasi-preinvex functions. Symmetry, 12.","DOI":"10.3390\/sym12010051"},{"key":"ref_25","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":"Aequ. Math."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Miao, L., Yang, W., and Zhang, X. (2010, January 10\u201312). Projection on convex set and its application in testing force closure properties of robotic grasping. Proceedings of the Intelligent Robotics and Applications\u2014Third International Conference, ICIRA 2010, Shanghai, China.","DOI":"10.1007\/978-3-642-16587-0_22"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"5783","DOI":"10.2298\/FIL1718783A","article-title":"On strongly generalized convex functions","volume":"31","author":"Awan","year":"2017","journal-title":"Filomat"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"15","DOI":"10.7494\/OpMath.2011.31.1.15","article-title":"On strongly midconvex functions","volume":"31","author":"Azocar","year":"2011","journal-title":"Opusc. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1023\/A:1024787424532","article-title":"Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints","volume":"118","author":"Lin","year":"2003","journal-title":"J. Optim. Theory Appl."},{"key":"ref_30","first-page":"111","article-title":"On strongly generalized convex functions of higher order","volume":"22","author":"Mishra","year":"2019","journal-title":"Math. Inequal. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Mohsen, B.B., Noor, M.A., Noor, K.I., and Postolache, M. (2019). Strongly convex functions of higher order involving bifunction. Mathematics, 7.","DOI":"10.3390\/math7111028"},{"key":"ref_32","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_33","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1016\/j.jmaa.2006.02.086","article-title":"On h-convexity","volume":"326","author":"Varosanec","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_34","first-page":"13","article-title":"Stetigkeitsaussagen fur eine Klasse verallgemeinerter konvexer funktionen in topologischen linearen Raumen","volume":"23","author":"Breckner","year":"1978","journal-title":"Publ. Inst. Math."},{"key":"ref_35","first-page":"335","article-title":"Some inequalities of Hadamard type","volume":"21","author":"Dragomir","year":"1995","journal-title":"Soochow J. Math."},{"key":"ref_36","unstructured":"Dragomir, S.S., and Pearce, C.E.M. (2003). Selected Topics on Hermite\u2013Hadamard Inequalities and Applications. Math. Preprint Arch., 463\u2013817."},{"key":"ref_37","unstructured":"Godunova, E.K., and Levin, V.I. (1985). Inequalities for functions of a broad class that contains convex, monotone and some other forms of functions. Numerical Mathematics and Mathematical Physics, Moskov. Gos. Ped. Inst."},{"key":"ref_38","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_39","doi-asserted-by":"crossref","first-page":"323","DOI":"10.4067\/S0716-09172015000400002","article-title":"Inequalities of Hermite\u2013Hadamard type for h-convex functions on linear spaces","volume":"34","author":"Dragomir","year":"2015","journal-title":"Proyecciones"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"173","DOI":"10.7153\/jmi-10-15","article-title":"On \u03a8-convex functions","volume":"10","author":"Gordji","year":"2016","journal-title":"J. Math. Inequal."},{"key":"ref_41","first-page":"112","article-title":"Inequalities via generalized h-convex functions","volume":"7","author":"Noor","year":"2018","journal-title":"Prob. Anal. Issues Anal."},{"key":"ref_42","first-page":"89","article-title":"On k-fractional integrals and application","volume":"7","author":"Mubeen","year":"2012","journal-title":"Int. J. Contemp. Math. Sci."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"269","DOI":"10.4153\/CMB-1976-042-4","article-title":"Weak parallelogram laws for Banach spaces","volume":"19","author":"Bynum","year":"1976","journal-title":"Can. Math. Bull."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1007\/s10998-014-0078-4","article-title":"Weak parallelogram laws on Banach spaces and applications to prediction","volume":"71","author":"Cheng","year":"2015","journal-title":"Period. Math. Hung."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1127","DOI":"10.1016\/0362-546X(91)90200-K","article-title":"Inequalities in Banach spaces with applications","volume":"16","author":"Xu","year":"1991","journal-title":"Nonlinear Anal. TMA"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/2\/222\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T08:54:01Z","timestamp":1760172841000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/2\/222"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,2]]},"references-count":45,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,2]]}},"alternative-id":["sym12020222"],"URL":"https:\/\/doi.org\/10.3390\/sym12020222","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,2]]}}}