Guessab, Allal; Schmeisser, Gerhard Sharp integral inequalities of the Hermite-Hadamard type. (English) Zbl 1012.26013 J. Approximation Theory 115, No. 2, 260-288 (2002). Authors’ summary: “We consider a family of two-point quadrature formulae and establish sharp estimates for the remainders under various regularity conditions. Improved forms of certain integral inequalities due to Hermite and Hadamard, Iyengar, Milovanović and Pečarić, and others are obtained as special cases. Our results can also be interpreted as analogues to a theorem of Ostrowski on the deviation of a function from its averages. Furthermore, we establish a generalization of a result of Fink concerning \(L^p\) estimates for the remainder of the trapezoidal rule and present the best constants in the error bounds”. Reviewer: George A.Anastassiou (Memphis) Cited in 2 ReviewsCited in 50 Documents MSC: 26D15 Inequalities for sums, series and integrals 65D30 Numerical integration 41A55 Approximate quadratures 65D32 Numerical quadrature and cubature formulas Keywords:Hermite-Hadamard inequality; two-point quadrature formulae; integral inequalities; trapezoidal rule PDF BibTeX XML Cite \textit{A. Guessab} and \textit{G. Schmeisser}, J. Approx. Theory 115, No. 2, 260--288 (2002; Zbl 1012.26013) Full Text: DOI OpenURL References: [1] Bojanov, B.D., Optimal methods of integration in the class of differentiable functions, Zastosowania mat., 15, 105-115, (1976) · Zbl 0353.65010 [2] Dragomir, S.S.; Cho, Y.J.; Kim, S.S., Inequalities of Hadamard’s type for Lipschitzian mappings and their applications, J. math. anal. appl., 245, 489-501, (2000) · Zbl 0956.26015 [3] Dragomir, S.S.; Pearce, C.E.M., Selected topics on hermite – hadamard inequalities and applications, (2000) · Zbl 0960.26004 [4] Fink, A.M., Bounds on the deviation of a function from its averages, Czechoslovak math. J., 42, 289-310, (1992) · Zbl 0780.26011 [5] Gradshteyn, I.S.; Ryzhik, I.M., () [6] Hörmander, L., Notions of convexity, (1994), Birkhäuser Boston · Zbl 0835.32001 [7] Iyengar, K.S.K., Note on an inequality, Math. student, 6, 75-76, (1938) · Zbl 0018.37105 [8] Mitrinović, D.S.; Pecarić, J.E.; Fink, A.M., Inequalities involving functions and their integrals and derivatives, (1991), Kluwer Academic Dordrecht · Zbl 0744.26011 [9] Ostrowski, A., Über die absolutabweichung einer differenzierbaren funktion von ihrem integralmittelwert, Comment. math. helv., 10, 226-227, (1938) · Zbl 0018.25105 [10] Rahman, Q.I.; Schmeisser, G., Analytic theory of polynomials, (2002), Oxford Univ. Press Oxford · Zbl 1072.30006 [11] Szegő, G., Orthogonal polynomials, (1975), Amer. Math. Soc Providence · JFM 65.0278.03 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.