Ostrowski Type Inequalities for Functions whose Derivatives are h-Convex in Absolute Value

Open access

Abstract

Some new inequalities of Ostrowski type for functions whose derivatives are h-convex in modulus are given. Applications for midpoint inequalities are provided as well.

[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.

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

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

[4] 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 [Online http://rgmia.org/papers/v5n2/Paperwapp2q.pdf].

[5] M. Bombardelli and S. Varosanec, Properties of h-convex functions related to the Hermite-Hadamard-Fejer inequalities. Comput. Math. Appl. 58 (2009), no. 9, 1869-1877.

[6] W. W. Breckner, Stetigkeitsaussagen fur eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Raumen. (German) Publ. Inst. Math. (Beograd) (N.S.) 23(37) (1978), 13-20.

[7] W. W. Breckner and G. Orban, Continuity properties of rationally s-convex mappings with values in an ordered topological linear space. Universitatea “Babes-Bolyai”, Facultatea de Matematica, Cluj-Napoca, 1978. viii+92 pp.

[8] 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.

[9] 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] 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.

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

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

[13] S. S. Dragomir, The Ostrowski’s integral inequality for Lipschitzian mappings and applications, Comp. Math. Appl., 38 (1999), 33-37.

[14] S. S. Dragomir, On the Ostrowski’s inequality for Riemann-Stieltjes integral, Korean J. Appl. Math., 7 (2000), 477-485.

[15] S. S. Dragomir, On the Ostrowski’s inequality for mappings of bounded variation and applications, Math. Ineq. & Appl., 4(1) (2001), 33-40.

[16] S. S. Dragomir, On the Ostrowski inequality for Riemann-Stieltjes integral ∫ab f (t) du (t) where f is of H¨older type and u is of bounded variation and applications, J. KSIAM, 5(1) (2001), 35-45.

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

[18] 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.

[19] S. S. Dragomir, An Ostrowski like inequality for convex functions and applications, Revista Math. Complutense, 16(2) (2003), 373-382.

[20] 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

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

[22] 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.

[23] 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.

[24] 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.

[25] 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.

[26] S. S. Dragomir and C. E. M. Pearce, Quasi-convex functions and Hadamard’s inequality, Bull. Austral. Math. Soc. 57 (1998), 377-385.

[27] S. S. Dragomir, J. Pecaric and L. Persson, Some inequalities of Hadamard type. Soochow J. Math. 21 (1995), no. 3, 335-341.

[28] S. S. Dragomir and Th. M. Rassias (Eds), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publisher, 2002.

[29] 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.

[30] 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.

[31] 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.

[32] 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

[33] A. M. Fink, Bounds on the deviation of a function from its averages, Czechoslovak Math. J., 42(117) (1992), No. 2, 298-310.

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

[35] U. S. Kirmaci, M. Klaricic Bakula, M. E Ozdemir and J. Pecaric, Hadamard-type inequalities for s-convex functions. Appl. Math. Comput. 193 (2007), no. 1, 26-35.

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

[37] D. S. Mitrinovic and J. E. Pecaric, Note on a class of functions of Godunova and Levin. C. R. Math. Rep. Acad. Sci. Canada 12 (1990), no. 1, 33-36.

[38] A. Ostrowski, Uber die Absolutabweichung einer differentienbaren Funktionen von ihren Inte- gralmittelwert, Comment. Math. Hel, 10 (1938), 226-227.

[39] 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.

[40] J. E. Pecaric 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. “Babes-Bolyai”, Cluj-Napoca, 1989.

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

[42] 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.

[43] E. Set, M. E. Ozdemir and M. Z. Sarikaya, 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.

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

[45] M. Tunc, Ostrowski-type inequalities via h-convex functions with applications to special means. J. Inequal. Appl. 2013, 2013:326.

[46] S. Varosanec, On h-convexity. J. Math. Anal. Appl. 326 (2007), no. 1, 303-311.