New inequalities of Hermite-Hadamard type for HA-convex functions

Authors

  • Sever Dragomir Victoria University, Australia

DOI:

https://doi.org/10.33993/jnaat471-1119

Keywords:

Convex functions, Integral inequalities, HA-Convex functions
Abstract views: 342

Abstract

Some new inequalities of Hermite-Hadamard type for HA-convex functions defined on positive intervals are given.

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References

G. D. Anderson, M.K. Vamanamurthy and M. Vuorinen, Generalized convexityand inequalities, J. Math. Anal. Appl.335(2007), 1294–1308, https://doi.org/10.1016/j.jmaa.2007.02.016. DOI: https://doi.org/10.1016/j.jmaa.2007.02.016

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. Inequality 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, https://rgmia.org/papers/v5n2/Paperwapp2q.pdf.

E.F. Beckenbach, Convex functions, Bull. Amer. Math. Soc., 54(1948), 439–460, https://doi.org/10.1090/S0002-9904-1948-08994-7. DOI: https://doi.org/10.1090/S0002-9904-1948-08994-7

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, https://doi.org/10.1016/j.camwa.2009.07.073 DOI: https://doi.org/10.1016/j.camwa.2009.07.073

W.W. Breckner, Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen.(German) Publ. Inst. Math. (Beograd) (N.S.) 23(37) (1978), pp. 13–20.

W.W. Breckner and G. Orbán, Continuity properties of rationally s-convex mappings with values in an ordered topological linear space. Universitatea ”Babe?-Bolyai”, Facultatea de Matematica, Cluj-Napoca, 1978. viii+92 pp.

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. DOI: https://doi.org/10.1201/9780429123610-4

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. DOI: https://doi.org/10.1142/9789812776372_0006

P. Cerone, S.S. Dragomir and J. Roumeliotis, Some Ostrowski type inequalities for time differentiable mappings and applications, Demonstratio Mathematica, 32(2) (1999), pp. 697-712. DOI: https://doi.org/10.1515/dema-1999-0404

G. Cristescu, Hadamard type inequalities for convolution of h convex functions. Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity 8 (2010), 3–11.

S.S. Dragomir, Some remarks on Hadamard’s inequalities for convex functions, Extracta Math., 9 (1994) no. 2, 88-94

S.S. Dragomir, Ostrowski’s inequality for monotonous mappings and applications, J. KSIAM 3(1) (1999), 127-135.

S.S. Dragomir, The Ostrowski’s integral inequality for Lipschitzian mappings and applications, Comp. Math. Appl., 38 (1999), 33-37, https://doi.org/10.1016/S0898-1221(99)00282-5 DOI: https://doi.org/10.1016/S0898-1221(99)00282-5

S.S. Dragomir, On the Ostrowski’s inequality for Riemann-Stieltjes integral, Korean J. Appl. Math., 7 (2000), 477-485. DOI: https://doi.org/10.1007/BF03012272

S.S. Dragomir, On the Ostrowski’s inequality for mappings of bounded variation and applications, Math. Ineq. & Appl. 4(1) (2001), 33-40, http://doi.org/10.7153/mia-04-05 DOI: https://doi.org/10.7153/mia-04-05

S.S. Dragomir, On the Ostrowski inequality for Riemann-Stieltjes integral ?_[a,b] f(t) du(t) where f is of Hölder type and u is of bounded variation and applications, J. KSIAM5(1) (2001), 35-45.

S.S. Dragomir, Ostrowski type inequalities for isotonic linear functionals, J. Inequal. Pure & Appl. Math. 3(5 (2002), Art. 68.

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.

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) 2, Article 31.

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) 3, Article 35.

S.S. Dragomir, An Ostrowski like inequality for convex functions and applications, Revista Math. Complutense, 16(2) (2003), 373–382. DOI: https://doi.org/10.5209/rev_REMA.2003.v16.n2.16807

S.S. Dragomir, Operator Inequalities of Ostrowski and Trapezoidal Type. Springer Briefs in Mathematics. Springer, New York, 2012. x+112 pp. ISBN: 978-1-4614-1778-1 DOI: https://doi.org/10.1007/978-1-4614-1779-8_3

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

S.S. Dragomir, Some new inequalities of Hermite-Hadamard type for GA-convex functions, Preprint RGMIA Res. Rep. Coll.18(2015), Art. 33. [http://rgmia.org/papers/v18/v18a33.pdf].

S.S.Dragomir, Inequalities of Hermite-Hadamard type for HA-convex functions, Preprint RGMIA Res. Rep. Coll. 18 (2015), Art.38. [http://rgmia.org/papers/v18/v18a38.pdf].

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) (1999) 4, 301–314.

S.S. Dragomir and S. Fitzpatrick, The Hadamard inequalities for s-convex functions in the second sense. Demonstratio Math., 32 (1999) 4, 687–696, https://doi.org/10.1515/dema-1999-0403 DOI: https://doi.org/10.1515/dema-1999-0403

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, https://doi.org/10.1515/dema-2000-0106 DOI: https://doi.org/10.1515/dema-2000-0106

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.

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, https://doi.org/10.1007/bf02093506 DOI: https://doi.org/10.1007/BF02093506

S.S. Dragomir, C.E.M. Pearce, Quasi-convex functions and Hadamard’s inequality, Bull. Austral. Math. Soc.,57 (1998), 377–385, https://doi.org/10.1017/S0004972700031786 DOI: https://doi.org/10.1017/S0004972700031786

S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000 [Online http://rgmia.org/monographs/hermitehadamard.html]

S.S. Dragomir, J. P?cari? and L. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21(1995) no. 3, 335–341.

S.S. Dragomir and Th.M. Rassias (Eds), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publisher, 2002. DOI: https://doi.org/10.1007/978-94-017-2519-4

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. Math.,28(1997), 239–244, https://doi.org/10.5556/j.tkjm.28.1997.4320 DOI: https://doi.org/10.5556/j.tkjm.28.1997.4320

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, https://doi.org/10.1016/S0893-9659(97)00142-0. DOI: https://doi.org/10.1016/S0893-9659(97)00142-0

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 (1998) no. 3, 245–304.

A. El Farissi, Simple proof and refeinment of Hermite-Hadamard inequality, J. Math. Ineq., 4 (2010) no. 3, 365-369, https://doi.org/10.7153/jmi-04-33. DOI: https://doi.org/10.7153/jmi-04-33

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,1985.

H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aeq. Math., 48 (1994) no. 1, 100–111, https://doi.org/10.1007/bf01837981. DOI: https://doi.org/10.1007/BF01837981

I. I?can, Hermite-Hadamard type inequalities for harmonically convex functions, Hacettepe J. Math. Stat., 43 (2014) 6, 935–942. DOI: https://doi.org/10.15672/HJMS.2014437519

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., 13 (2010) no. 1, 1–32. DOI: https://doi.org/10.7153/mia-13-01

U.S. Kirmaci, M. Klari?i? Bakula, M.E. Özdemir and J. Pe?ari?, Hadamard-type inequalities for s-convex functions, Appl. Math. Comput., 193 (2007) no. 1, 26–35, https://doi.org/10.1016/j.amc.2007.03.030 DOI: https://doi.org/10.1016/j.amc.2007.03.030

M.A. Latif, On some inequalities for h-convex functions, Int. J. Math. Anal. (Ruse), 4 (2010) no. 29–32, 1473-1482.

D.S. Mitrinovi? and I.B. Lackovi?, Hermite and convexity, Aeq. Math., 28(1985), 229–232, https://doi.org/10.1007/bf02189414 DOI: https://doi.org/10.1007/BF02189414

D.S. Mitrinovi?c and J.E. Pe?cari?, Note on a class of functions of Godunova and Levin, C.R. Math. Rep. Acad. Sci. Canada, 12(1990) no. 1, 33–36.

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. DOI: https://doi.org/10.37193/CMI.2014.01.14

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, https://doi.org/10.1006/jmaa.1999.6593 DOI: https://doi.org/10.1006/jmaa.1999.6593

J.E. Pe?ari? 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.“Babe?-Bolyai”, Cluj-Napoca, 1989.

J. Pe?ari? and S. Dragomir, A generalization of Hadamard’s inequality for isotonic linear functionals, Radovi Mat. (Sarajevo), 7 (1991), 103–107.

M. Radulescu, S. Radulescu and P. Alexandrescu, On the Godunova-Levin-Schurclass of functions, Math. Inequal. Appl.,12(2009) no. 4, 853–862, https://doi.org//10.7153/mia-12-69. DOI: https://doi.org/10.7153/mia-12-69

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, dx.doi.org/10.7153/jmi-02-30 DOI: https://doi.org/10.7153/jmi-02-30

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.

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.

M. Tunç, Ostrowski-type inequalities via h-convex functions with applications to special means, J. Inequal. Appl., 2013, 2013:326, https://doi.org/10.1186/1029-242X-2013-326. DOI: https://doi.org/10.1186/1029-242X-2013-326

S. Varošanec, On h-convexity, J. Math. Anal. Appl., 326 (2007) no. 1, 303–311, https://doi.org/10.1016/j.jmaa.2006.02.086 DOI: https://doi.org/10.1016/j.jmaa.2006.02.086

X.-M. Zhang, Y.-M. ChuandX.-H. Zhang, The Hermite-Hadamard type inequality of GA-convex functions and its application, J. Ineq. Appl., 2010, Article ID 507560, 11pp, https://doi.org/10.1155/2010/507560. DOI: https://doi.org/10.1155/2010/507560

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Published

2018-08-06

How to Cite

Dragomir, S. (2018). New inequalities of Hermite-Hadamard type for HA-convex functions. J. Numer. Anal. Approx. Theory, 47(1), 26–41. https://doi.org/10.33993/jnaat471-1119

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