×

More about Hermite-Hadamard inequalities, Cauchy’s means, and superquadracity. (English) Zbl 1205.26028

Summary: New results associated with Hermite-Hadamard inequalities for superquadratic functions are given. A set of Cauchy’s type means is derived from these Hermite-Hadamard-type inequalities, and its log-convexity and monotonicity are proved.

MSC:

26D15 Inequalities for sums, series and integrals
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Pečarić JE, Proschan F, Tong YL: Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering. Volume 187. Academic Press, Boston, Mass, USA; 1992:xiv+467. · Zbl 0749.26004
[2] Dragomir, SS, On Hadamard’s inequalities for convex functions, Mathematica Balkanica, 6, 215-222, (1992) · Zbl 0834.26010
[3] Dragomir SS, McAndrew A: Refinements of the Hermite-Hadamard inequality for convex functions. Journal of Inequalities in Pure and Applied Mathematics 2005., 6(5, article 140): · Zbl 1085.26012
[4] Banić, S, Mappings connected with Hermite-Hadamard inequalities for superquadratic functions, Journal of Mathematical Inequalities, 3, 577-589, (2009) · Zbl 1182.26043
[5] Dragomir, SS, A mapping in connection to Hadamard’s inequalities, Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse, 128, 17-20, (1991) · Zbl 0747.26015
[6] Dragomir, SS, Two mappings in connection to Hadamard’s inequalities, Journal of Mathematical Analysis and Applications, 167, 49-56, (1992) · Zbl 0758.26014
[7] Abramovich, S; Jameson, G; Sinnamon, G, Refining Jensen’s inequality, Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, 47, 3-14, (2004) · Zbl 1150.26333
[8] Banić, S; Varošanec, S, Functional inequalities for superquadratic functions, International Journal of Pure and Applied Mathematics, 43, 537-549, (2008) · Zbl 1154.39027
[9] Anwar M, Jakšetić J, Pečarić J, Atiq Ur Rehman : Exponential convexity, positive semi-definite matrices and fundamental inequalities. accepted in Journal of Mathematical Inequalities accepted in Journal of Mathematical Inequalities · Zbl 1218.26007
[10] Hirschman II, Widder DV: The Convolution Transform. Princeton University Press, Princeton, NJ, USA; 1955:x+268.
[11] Sierpinski, W, A family of the Cauchy type Mean-value theorems, Fundamenta Mathematicae, 1, 125-129, (1920)
[12] Abramovich S, Farid G, Pečarić J: More About Jensen’s Inequality and Cauchy’s Means for Superquadratic Functions. submitted submitted · Zbl 1264.26019
[13] Anwar, M; Latif, N; Pečarić, J, Cauchy means of the Popoviciu type, No. 2009, 18-16, (2009) · Zbl 1175.26033
[14] Anwar, M; Pečarić, J, Cauchy’s means of Levinson type, Journal of Inequalities in Pure and Applied Mathematics, 9, 8, (2008) · Zbl 1163.26314
[15] Anwar, M; Pečarić, J, New means of Cauchy’s type, No. 2008, 15-10, (2008) · Zbl 1148.26035
[16] Pečarić, JE; Perić, I; Srivastava, HM, A family of the Cauchy type Mean-value theorems, Journal of Mathematical Analysis and Applications, 306, 730-739, (2005) · Zbl 1068.26008
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.