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J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1006\/jmaa.2000.6769","article-title":"Inequalities of Hadamard\u2019s type for Lipschitzian mappings and their applications","volume":"245","author":"Dragomir","year":"2000","journal-title":"J. Math. Anal. Appl."},{"key":"ref_4","unstructured":"Dragomir, S.S., and Pearce, C.E.M. (2021, July 10). Selected Topics on Hermite-Hadamard Inequalities and Applications; RGMIA Monographs. Available online: http:\/\/rgmia.org\/monographs\/hermite_hadamard.html."},{"key":"ref_5","first-page":"245","article-title":"Refinement of Hermite-Hadamard inequality","volume":"68","author":"Feng","year":"2012","journal-title":"Far East J. Math. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"587","DOI":"10.7153\/jmi-04-52","article-title":"A note on the Hermite-Hadamard inequality","volume":"4","author":"Gao","year":"2010","journal-title":"J. Math. 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J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/s006050200015","article-title":"Univariate Ostrowski inequalities, revisited","volume":"135","author":"Anastassiou","year":"2002","journal-title":"Monatsh. Math."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Anastassiou, G.A. (2019). Midpoint-type rules from an inequalities point of view. Handbook of Analytic-Computational Methods in Applied Mathematics, CRC Press.","DOI":"10.1201\/9780429123610"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Gulati, C. (2002). New bounds for the three-point rule involving the Riemann-Stieltjes integrals. Advances in Statistics Combinatorics and Related Areas, World Science Publishing.","DOI":"10.1142\/5134"},{"key":"ref_20","first-page":"697","article-title":"Some Ostrowski type inequalities for n-time differentiable mappings and applications","volume":"32","author":"Cerone","year":"1999","journal-title":"Demonstr. Math."},{"key":"ref_21","first-page":"1623","article-title":"A note concerning several Hermite-Hadamard inequalities for different types of convex functions","volume":"6","author":"Ciurdariu","year":"2012","journal-title":"Int. J. Math. Anal."},{"key":"ref_22","first-page":"19","article-title":"Weighted norm inequalities of trapezoid type for Frechet differentiable functions in Banach spaces","volume":"23","author":"Dragomir","year":"2020","journal-title":"RGMIA Res. Rep. Coll."},{"key":"ref_23","first-page":"61","article-title":"Inequalities of Lipschitz type for power series in Banach algebras","volume":"29","author":"Dragomir","year":"2015","journal-title":"Ann. Math. 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