Kirmaci, Uǧur S. Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. (English) Zbl 1034.26019 Appl. Math. Comput. 147, No. 1, 137-146 (2004). The author proves certain inequalities of Hadamard type for convex mappings. We quote the following result: Let \(| f'|\) be convex on \([a,b]\). Then \[ \Biggl| M_f(a, b)- f\Biggl({a+b\over 2}\Biggr)\Biggr|\leq {b-a\over 8} (| f'(a)|+| f'(b)|) \] (\(M_f(a,b)\) denotes the integral means of \(f\) on \([a,b]\)). Some applications for special means of two arguments are also pointed out. However, these results are not compared to the existing relations in the theory of means. Reviewer: József Sándor (Cluj-Napoca) Cited in 2 ReviewsCited in 147 Documents MSC: 26D15 Inequalities for sums, series and integrals 26A51 Convexity of real functions in one variable, generalizations Keywords:Hermite-Hadamard inequality; convex functions; special means; midpoint formula PDFBibTeX XMLCite \textit{U. S. Kirmaci}, Appl. Math. Comput. 147, No. 1, 137--146 (2004; Zbl 1034.26019) Full Text: DOI References: [1] Pečarić, J. E.; Proschan, F.; Tong, Y. L., Convex Functions, Partial Ordering and Statistical Applications (1991), Academic Press: Academic Press New York [2] Dragomir, S. S.; Agarwal, R. P., Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11, 5, 91-95 (1998) · Zbl 0938.26012 [3] Özdemir, M. E., A theorem on mappings with bounded derivatives with applications to quadrature rules and means, Appl. Math. Comput., 138, 425-434 (2002) · Zbl 1033.26023 [4] Dragomir, S. S., On Hadamard’s inequalities for convex functions, Math. Balkanica, 6, 215-222 (1992) · Zbl 0834.26010 [5] Dragomir, S. S., Two mappings in connection to Hadamard’s inequality, J. Math. Anal. Appl., 167, 49-56 (1992) · Zbl 0758.26014 [6] Pearce, C. E.M.; Pecaric, J., Inequalities for differentiable mappings with application to special means and quadrature formula, Appl. Math. Lett., 13, 51-55 (2000) · Zbl 0970.26016 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.