Inequalities of Hermite-Hadamard Type for HA-Convex Functions

[1] M. Alomari and M. Darus, The Hadamard's inequality for s-convex function. Int. J. Math. Anal. (Ruse) 2 (2008), no. 13-16, 639-646.Search in Google Scholar

[2] M. Alomari and M. Darus, Hadamard-type inequalities for s-convex functions. Int. Math. Forum 3 (2008), no. 37-40, 1965-1975.Search in Google Scholar

[3] G. A. Anastassiou, Univariate Ostrowski inequalities, revisited. Monatsh. Math., 135 (2002), no. 3, 175-189.Search in Google Scholar

[4] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Generalized convexity and inequalities, J. Math. Anal. Appl. 335 (2007) 1294-1308.10.1016/j.jmaa.2007.02.016Search in Google Scholar

[5] N. S. Barnett, P. Cerone, S. S. Dragomir, M. R. Pinheiro,and A. Sofo, Ostrowski type inequalities for functions whose modulus of the derivatives are convex and applications. In- equality Theory and Applications, Vol. 2 (Chinju/Masan, 2001), 19-32, Nova Sci. Publ., Hauppauge, NY, 2003. Preprint: RGMIA Res. Rep. Coll. 5 (2002), No. 2, Art. 1 [Online http://rgmia.org/papers/v5n2/Paperwapp2q.pdf].Search in Google Scholar

[6] E. F. Beckenbach, Convex functions, Bull. Amer. Math. Soc. 54(1948), 439-460.10.1090/S0002-9904-1948-08994-7Search in Google Scholar

[7] M. Bombardelli and S. Varošanec, Properties of h-convex functions related to the Hermite-Hadamard-Fejér inequalities. Comput. Math. Appl. 58 (2009), no. 9, 1869-1877.Search in Google Scholar

[8] W. W. Breckner, Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen. (German) Publ. Inst. Math. (Beograd) (N.S.) 23(37) (1978), 13-20.Search in Google Scholar

[9] W. W. Breckner and G. Orbán, Continuity properties of rationally s-convex mappings with values in an ordered topological linear space. Universitatea "Babeș-Bolyai", Facultatea de Matematica, Cluj-Napoca, 1978. viii+92 pp.Search in Google Scholar

[10] P. Cerone and S. S. Dragomir, Midpoint-type rules from an inequalities point of view, Ed. G. A. Anastassiou, Handbook of Analytic-Computational Methods in Applied Mathematics, CRC Press, New York. 135-200.10.1201/9780429123610-4Search in Google Scholar

[11] P. Cerone and S. S. Dragomir, New bounds for the three-point rule involving the Riemann-Stieltjes integrals, in Advances in Statistics Combinatorics and Related Areas, C. Gulati, et al. (Eds.), World Science Publishing, 2002, 53-62.10.1142/9789812776372_0006Search in Google Scholar

[12] P. Cerone, S. S. Dragomir and J. Roumeliotis, Some Ostrowski type inequalities for n-time differentiable mappings and applications, Demonstratio Mathematica, 32(2) (1999), 697|712.10.1515/dema-1999-0404Search in Google Scholar

[13] G. Cristescu, Hadamard type inequalities for convolution of h-convex functions. Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity 8 (2010), 3-11.Search in Google Scholar

[14] S. S. Dragomir, Ostrowski's inequality for monotonous mappings and applications, J. KSIAM, 3(1) (1999), 127-135.Search in Google Scholar

[15] S. S. Dragomir, The Ostrowski's integral inequality for Lipschitzian mappings and applications, Comp. Math. Appl., 38 (1999), 33-37.10.1016/S0898-1221(99)00282-5Search in Google Scholar

[16] S. S. Dragomir, On the Ostrowski's inequality for Riemann-Stieltjes integral, Korean J. Appl. Math., 7 (2000), 477-485.10.1007/BF03012263Search in Google Scholar

[17] S. S. Dragomir, On the Ostrowski's inequality for mappings of bounded variation and applications, Math. Ineq. & Appl., 4(1) (2001), 33-40.10.7153/mia-04-05Search in Google Scholar

[18] S. S. Dragomir, On the Ostrowski inequality for Riemann-Stieltjes integral R b a f (t) du (t) where f is of Hölder type and u is of bounded variation and applications, J. KSIAM, 5(1) (2001), 35-45.Search in Google Scholar

[19] S. S. Dragomir, Ostrowski type inequalities for isotonic linear functionals, J. Inequal. Pure & Appl. Math., 3(5) (2002), Art. 68.Search in Google Scholar

[20] S. S. Dragomir, An inequality improving the first Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products. J. Inequal. Pure Appl. Math. 3 (2002), no. 2, Article 31, 8 pp.Search in Google Scholar

[21] S. S. Dragomir, An inequality improving the first Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math. 3 (2002), No. 2, Article 31.Search in Google Scholar

[22] S. S. Dragomir, An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math. 3 (2002), No.3, Article 35.Search in Google Scholar

[23] S. S. Dragomir, An Ostrowski like inequality for convex functions and applications, Revista Math. Complutense, 16(2) (2003), 373-382.10.5209/rev_REMA.2003.v16.n2.16807Search in Google Scholar

[24] S. S. Dragomir, Operator Inequalities of Ostrowski and Trapezoidal Type. Springer Briefs in Mathematics. Springer, New York, 2012. x+112 pp. ISBN: 978-1-4614-1778-110.1007/978-1-4614-1779-8_3Search in Google Scholar

[25] S. S. Dragomir, Some new inequalities of Hermite-Hadamard type for GAconvex functions, Preprint RGMIA Res. Rep. Coll. 18 (2015), Art. 30. [http://rgmia.org/papers/v18/v18a30.pdf].Search in Google Scholar

[26] S. S. Dragomir, P. Cerone, J. Roumeliotis and S.Wang, A weighted version of Ostrowski inequality for mappings of Hölder type and applications in numerical analysis, Bull. Math. Soc. Sci. Math. Romanie, 42(90) (4) (1999), 301-314.Search in Google Scholar

[27] S.S. Dragomir and S. Fitzpatrick, The Hadamard inequalities for s-convex functions in the second sense. Demonstratio Math. 32 (1999), no. 4, 687-696.Search in Google Scholar

[28] S.S. Dragomir and S. Fitzpatrick,The Jensen inequality for s-Breckner convex functions in linear spaces. Demonstratio Math. 33 (2000), no. 1, 43-49.Search in Google Scholar

[29] S. S. Dragomir and B. Mond, On Hadamard's inequality for a class of functions of Godunova and Levin. Indian J. Math. 39 (1997), no. 1, 1-9.Search in Google Scholar

[30] S. S. Dragomir and C. E. M. Pearce, On Jensen's inequality for a class of functions of Godunova and Levin. Period. Math. Hungar. 33 (1996), no. 2, 93-100.Search in Google Scholar

[31] S. S. Dragomir and C. E. M. Pearce, Quasi-convex functions and Hadamard's inequality, Bull. Austral. Math. Soc. 57 (1998), 377-385.10.1017/S0004972700031786Search in Google Scholar

[32] S. S. Dragomir, J. Pečarić and L. Persson, Some inequalities of Hadamard type. Soochow J. Math. 21 (1995), no. 3, 335-341.Search in Google Scholar

[33] S. S. Dragomir and Th. M. Rassias (Eds), Ostrowski Type Inequalities and Applications in Nu- merical Integration, Kluwer Academic Publisher, 2002.10.1007/978-94-017-2519-4Search in Google Scholar

[34] S. S. Dragomir and S. Wang, A new inequality of Ostrowski's type in L1-norm and applications to some special means and to some numerical quadrature rules, Tamkang J. of Math., 28 (1997), 239-244.10.5556/j.tkjm.28.1997.4320Search in Google Scholar

[35] S. S. Dragomir and S. Wang, Applications of Ostrowski's inequality to the estimation of error bounds for some special means and some numerical quadrature rules, Appl. Math. Lett., 11 (1998), 105-109.10.1016/S0893-9659(97)00142-0Search in Google Scholar

[36] S. S. Dragomir and S. Wang, A new inequality of Ostrowski's type in Lp-norm and applications to some special means and to some numerical quadrature rules, Indian J. of Math., 40(3) (1998), 245-304.Search in Google Scholar

[37] A. El Farissi, Simple proof and refeinment of Hermite-Hadamard inequality, J. Math. Ineq. 4 (2010), No. 3, 365-369.Search in Google Scholar

[38] E. K. Godunova and V. I. Levin, Inequalities for functions of a broad class that contains convex, monotone and some other forms of functions. (Russian) Numerical mathematics and mathematical physics (Russian), 138-142, 166, Moskov. Gos. Ped. Inst., Moscow, 1985Search in Google Scholar

[39] H. Hudzik and L. Maligranda, Some remarks on s-convex functions. Aequationes Math. 48 (1994), no. 1, 100-111.Search in Google Scholar

[40] I. Ișcan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacettepe Jour- nal of Mathematics and Statistics, 43 (6) (2014), 935 - 942.10.15672/HJMS.2014437519Search in Google Scholar

[41] E. Kikianty and S. S. Dragomir, Hermite-Hadamard's inequality and the p-HH-norm on the Cartesian product of two copies of a normed space, Math. Inequal. Appl. (in press)Search in Google Scholar

[42] U. S. Kirmaci, M. Klaričić Bakula, M. E Özdemir and J. Pečarić, Hadamard-type inequalities for s-convex functions. Appl. Math. Comput. 193 (2007), no. 1, 26-35.Search in Google Scholar

[43] M. A. Latif, On some inequalities for h-convex functions. Int. J. Math. Anal. (Ruse) 4 (2010), no. 29-32, 1473-1482.Search in Google Scholar

[44] D. S. Mitrinović and I. B. Lacković, Hermite and convexity, Aequationes Math. 28 (1985), 229-232.10.1007/BF02189414Search in Google Scholar

[45] D. S. Mitrinović and J. E. Pečarić, Note on a class of functions of Godunova and Levin. C. R. Math. Rep. Acad. Sci. Canada 12 (1990), no. 1, 33-36.Search in Google Scholar

[46] M. A. Noor, K. I. Noor and M. U. Awan, Some inequalities for geometrically-arithmetically h-convex functions, Creat. Math. Inform. 23 (2014), No. 1, 91 - 98.Search in Google Scholar

[47] C. E. M. Pearce and A. M. Rubinov, P-functions, quasi-convex functions, and Hadamard-type inequalities. J. Math. Anal. Appl. 240 (1999), no. 1, 92-104.Search in Google Scholar

[48] J. E. Pečarić and S. S. Dragomir, On an inequality of Godunova-Levin and some refinements of Jensen integral inequality. Itinerant Seminar on Functional Equations, Approximation and Convexity (Cluj-Napoca, 1989), 263-268, Preprint, 89-6, Univ. "Babeș-Bolyai", Cluj-Napoca, 1989.Search in Google Scholar

[49] J. Pečarić and S. S. Dragomir, A generalization of Hadamard's inequality for isotonic linear functionals, Radovi Mat. (Sarajevo) 7 (1991), 103-107.Search in Google Scholar

[50] M. Radulescu, S. Radulescu and P. Alexandrescu, On the Godunova-Levin-Schur class of functions. Math. Inequal. Appl. 12 (2009), no. 4, 853-862.Search in Google Scholar

[51] M. Z. Sarikaya, A. Saglam, and H. Yildirim, On some Hadamard-type inequalities for h-convex functions. J. Math. Inequal. 2 (2008), no. 3, 335-341.Search in Google Scholar

[52] E. Set, M. E. Özdemir and M. Z. Sar_kaya, New inequalities of Ostrowski's type for s-convex functions in the second sense with applications. Facta Univ. Ser. Math. Inform. 27 (2012), no. 1, 67-82.Search in Google Scholar

[53] M. Z. Sarikaya, E. Set and M. E. Özdemir, On some new inequalities of Hadamard type involving h-convex functions. Acta Math. Univ. Comenian. (N.S.) 79 (2010), no. 2, 265-272.Search in Google Scholar

[54] M. Tunć, Ostrowski-type inequalities via h-convex functions with applications to special means. J. Inequal. Appl. 2013, 2013:326.10.1186/1029-242X-2013-326Search in Google Scholar

[55] S. Varošanec, On h-convexity. J. Math. Anal. Appl. 326 (2007), no. 1, 303-311.Search in Google Scholar

[56] X.-M. Zhang, Y.-M. Chu and X.-H. Zhang, The Hermite-Hadamard type inequality of GA-convex functions and its application, Journal of Inequalities and Applications, Volume 2010, Article ID 507560, 11 pages.10.1155/2010/507560Search in Google Scholar