Inequalities of Hermite-Hadamard Type for h-Convex Functions on Linear Spaces
DOI:
https://doi.org/10.4067/S0716-09172015000400002Keywords:
Convex functions, Integral inequalities, h-Convex functions.Abstract
Some inequalities of Hermite-Hadamard type for h-convex functions defined on convex subsets in real or complex linear spaces are given. Applications for norm inequalities are provided as well.References
[1] M. Alomari and M. Darus, The Hadamard’s inequality for s-convex function. Int. J. Math. Anal. (Ruse) 2, No. 13-16, pp. 639—646, (2008).
[2] M. Alomari and M. Darus, Hadamard-type inequalities for s-convex functions. Int. Math. Forum 3, No. 37-40, pp. 1965—1975, (2008).
[3] G. A. Anastassiou, Univariate Ostrowski inequalities, revisited. Monatsh. Math., 135, No. 3, pp. 175—189, (2002).
[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] E. F. Beckenbach, Convex functions, Bull. Amer. Math. Soc. 54, pp. 439—460, (1948).
[6] M. Bombardelli and S. Varosanec, Properties of h-convex functions related to the Hermite-Hadamard-Fejér inequalities. Comput. Math. Appl. 58, No. 9, pp. 1869—1877, (2009).
[7] W. W. Breckner, Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen. (German) Publ. Inst. Math. (Beograd) (N.S.) 23(37), pp. 13—20, (1978).
[8] W. W. Breckner and G. Orbán, Continuity properties of rationally s-convex mappings with values in an ordered topological linear space. Universitatea ”Babe¸s-Bolyai, Facultatea de Matematica, Cluj-Napoca, (1978). viii+92.
[9] 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] 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, pp. 53-62, (2002).
[11] P. Cerone, S. S. Dragomir and J. Roumeliotis, Some Ostrowski type inequalities for n-time differentiable mappings and applications, Demonstratio Mathematica, 32(2), pp. 697-712, (1999).
[12] G. Cristescu, Hadamard type inequalities for convolution of h-convex functions. Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity 8, pp. 3—11, (2010).
[13] S. S. Dragomir, Ostrowski’s inequality for monotonous mappings and applications, J. KSIAM, 3(1), pp. 127-135, (1999).
[14] S. S. Dragomir, The Ostrowski’s integral inequality for Lipschitzian mappings and applications, Comp. Math. Appl., 38, pp. 33-37, (1999).
[15] S. S. Dragomir, On the Ostrowski’s inequality for Riemann-Stieltjes integral, Korean J. Appl. Math., 7, pp. 477-485, (2000).
[16] S. S. Dragomir, On the Ostrowski’s inequality for mappings of bounded variation and applications, Math. Ineq. & Appl., 4 (1), pp. 33-40, (2001).
[17] S. S. Dragomir, On the Ostrowski inequality for Riemann-Stieltjes integral R b a f (t) du (t) where f is of H¨older type and u is of bounded variation and applications, J. KSIAM, 5 (1), pp. 35-45, (2001).
[18] S. S. Dragomir, Ostrowski type inequalities for isotonic linear functionals, J. Inequal. Pure & Appl. Math., 3(5) (2002), Art. 68.
[19] 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.
[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.
[21] S. S. Dragomir, An inequality improving the second HermiteHadamard 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.
[22] S. S. Dragomir, An Ostrowski like inequality for convex functions and applications, Revista Math. Complutense, 16 (2), pp. 373-382, (2003).
[23] 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.
[24] S. S. Dragomir, P. Cerone, J. Roumeliotis and S. Wang, A weighted version of Ostrowski inequality for mappings of H¨older type and applications in numerical analysis, Bull. Math. Soc. Sci. Math. Romanie, 42 (90) (4), pp. 301-314, (1999).
[25] S.S. Dragomir and S. Fitzpatrick, The Hadamard inequalities for sconvex functions in the second sense. Demonstratio Math. 32, No. 4, pp. 687—696, (1999).
[26] S.S. Dragomir and S. Fitzpatrick, The Jensen inequality for s-Breckner convex functions in linear spaces. Demonstratio Math. 33, No. 1, pp. 43—49, (2000).
[27] S. S. Dragomir and B. Mond, On Hadamard’s inequality for a class of functions of Godunova and Levin. Indian J. Math. 39, No. 1, pp. 1—9, (1997).
[28] 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, No. 2, pp. 93—100, (1996).
[29] S. S. Dragomir and C. E. M. Pearce, Quasi-convex functions and Hadamard’s inequality, Bull. Austral. Math. Soc. 57, pp. 377-385, (1998).
[30] S. S. Dragomir, J. Pecari´c and L. Persson, Some inequalities of Hadamard type. Soochow J. Math. 21, No. 3, pp. 335—341, (1995).
[31] S. S. Dragomir and Th. M. Rassias (Eds), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publisher, (2002).
[32] S. S. Dragomir and S. Wang, A new inequality of Ostrowski’s type in Li-norm and applications to some special means and to some numerical quadrature rules, Tamkang J. of Math., 28, pp. 239-244, (1997).
[33] 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 , pp. 105-109, (1998).
[34] 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), pp. 245-304, (1998).
[35] A. El Farissi, Simple proof and refeinment of Hermite-Hadamard inequality, J. Math. Ineq. 4, No. 3, pp. 365—369, (2010).
[36] 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), pp. 138—142, 166, Moskov. Gos. Ped. Inst., Moscow, (1985).
[37] H. Hudzik and L. Maligranda, Some remarks on s-convex functions. Aequationes Math. 48, No. 1, 100—111, (1994).
[38] 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)
[39] U. S. Kirmaci, M. Klarici´c Bakula, M. E Ozdemir and J. Pecari´c, Hadamard-type inequalities for s-convex functions. Appl. Math. Comput. 193, No. 1, pp. 26—35, (2007).
[40] M. A. Latif, On some inequalities for h-convex functions. Int. J. Math. Anal. (Ruse) 4, No. 29-32, pp. 1473—1482, (2010).
[41] G. Maksa and Z. Pales, The equality case in some recent convexity inequalities, Opuscula Math. 31, 2, pp. 269-277, (2011).
[42] D. S. Mitrinovic and I. B. Lackovic, Hermite and convexity, Aequationes Math. 28, pp. 229—232, (1985).
[43] 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, No. 1, pp. 33—36, (1990).
[44] C. E. M. Pearce and A. M. Rubinov, P-functions, quasi-convex functions, and Hadamard-type inequalities. J. Math. Anal. Appl. 240, No. 1, pp. 92—104, (1999).
[45] 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. ”Babe¸s-Bolyai, Cluj-Napoca, (1989).
[46] J. Pecaric and S. S. Dragomir, A generalization of Hadamard’s inequality for isotonic linear functionals, Radovi Mat. (Sarajevo) 7, pp. 103—107, (1991).
[47] M. Radulescu, S. Radulescu and P. Alexandrescu, On the GodunovaLevin-Schur class of functions. Math. Inequal. Appl. 12, No. 4, pp. 853—862, (2009).
[48] M. Z. Sarikaya, A. Saglam, and H. Yildirim, On some Hadamard-type inequalities for h-convex functions. J. Math. Inequal. 2, No. 3, pp. 335—341, (2008).
[49] 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, No. 1, pp. 67—82, (2012).
[50] 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, pp. 265-272, No. 2, (2010).
[51] M. Tun¸c, Ostrowski-type inequalities via h-convex functions with applications to special means. J. Inequal. Appl., 2013, 2013:326. [52] S. Varosanec, On h-convexity. J. Math. Anal. Appl. 326, No. 1, pp. 303-311, (2007).
[2] M. Alomari and M. Darus, Hadamard-type inequalities for s-convex functions. Int. Math. Forum 3, No. 37-40, pp. 1965—1975, (2008).
[3] G. A. Anastassiou, Univariate Ostrowski inequalities, revisited. Monatsh. Math., 135, No. 3, pp. 175—189, (2002).
[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] E. F. Beckenbach, Convex functions, Bull. Amer. Math. Soc. 54, pp. 439—460, (1948).
[6] M. Bombardelli and S. Varosanec, Properties of h-convex functions related to the Hermite-Hadamard-Fejér inequalities. Comput. Math. Appl. 58, No. 9, pp. 1869—1877, (2009).
[7] W. W. Breckner, Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen. (German) Publ. Inst. Math. (Beograd) (N.S.) 23(37), pp. 13—20, (1978).
[8] W. W. Breckner and G. Orbán, Continuity properties of rationally s-convex mappings with values in an ordered topological linear space. Universitatea ”Babe¸s-Bolyai, Facultatea de Matematica, Cluj-Napoca, (1978). viii+92.
[9] 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] 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, pp. 53-62, (2002).
[11] P. Cerone, S. S. Dragomir and J. Roumeliotis, Some Ostrowski type inequalities for n-time differentiable mappings and applications, Demonstratio Mathematica, 32(2), pp. 697-712, (1999).
[12] G. Cristescu, Hadamard type inequalities for convolution of h-convex functions. Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity 8, pp. 3—11, (2010).
[13] S. S. Dragomir, Ostrowski’s inequality for monotonous mappings and applications, J. KSIAM, 3(1), pp. 127-135, (1999).
[14] S. S. Dragomir, The Ostrowski’s integral inequality for Lipschitzian mappings and applications, Comp. Math. Appl., 38, pp. 33-37, (1999).
[15] S. S. Dragomir, On the Ostrowski’s inequality for Riemann-Stieltjes integral, Korean J. Appl. Math., 7, pp. 477-485, (2000).
[16] S. S. Dragomir, On the Ostrowski’s inequality for mappings of bounded variation and applications, Math. Ineq. & Appl., 4 (1), pp. 33-40, (2001).
[17] S. S. Dragomir, On the Ostrowski inequality for Riemann-Stieltjes integral R b a f (t) du (t) where f is of H¨older type and u is of bounded variation and applications, J. KSIAM, 5 (1), pp. 35-45, (2001).
[18] S. S. Dragomir, Ostrowski type inequalities for isotonic linear functionals, J. Inequal. Pure & Appl. Math., 3(5) (2002), Art. 68.
[19] 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.
[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.
[21] S. S. Dragomir, An inequality improving the second HermiteHadamard 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.
[22] S. S. Dragomir, An Ostrowski like inequality for convex functions and applications, Revista Math. Complutense, 16 (2), pp. 373-382, (2003).
[23] 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.
[24] S. S. Dragomir, P. Cerone, J. Roumeliotis and S. Wang, A weighted version of Ostrowski inequality for mappings of H¨older type and applications in numerical analysis, Bull. Math. Soc. Sci. Math. Romanie, 42 (90) (4), pp. 301-314, (1999).
[25] S.S. Dragomir and S. Fitzpatrick, The Hadamard inequalities for sconvex functions in the second sense. Demonstratio Math. 32, No. 4, pp. 687—696, (1999).
[26] S.S. Dragomir and S. Fitzpatrick, The Jensen inequality for s-Breckner convex functions in linear spaces. Demonstratio Math. 33, No. 1, pp. 43—49, (2000).
[27] S. S. Dragomir and B. Mond, On Hadamard’s inequality for a class of functions of Godunova and Levin. Indian J. Math. 39, No. 1, pp. 1—9, (1997).
[28] 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, No. 2, pp. 93—100, (1996).
[29] S. S. Dragomir and C. E. M. Pearce, Quasi-convex functions and Hadamard’s inequality, Bull. Austral. Math. Soc. 57, pp. 377-385, (1998).
[30] S. S. Dragomir, J. Pecari´c and L. Persson, Some inequalities of Hadamard type. Soochow J. Math. 21, No. 3, pp. 335—341, (1995).
[31] S. S. Dragomir and Th. M. Rassias (Eds), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publisher, (2002).
[32] S. S. Dragomir and S. Wang, A new inequality of Ostrowski’s type in Li-norm and applications to some special means and to some numerical quadrature rules, Tamkang J. of Math., 28, pp. 239-244, (1997).
[33] 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 , pp. 105-109, (1998).
[34] 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), pp. 245-304, (1998).
[35] A. El Farissi, Simple proof and refeinment of Hermite-Hadamard inequality, J. Math. Ineq. 4, No. 3, pp. 365—369, (2010).
[36] 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), pp. 138—142, 166, Moskov. Gos. Ped. Inst., Moscow, (1985).
[37] H. Hudzik and L. Maligranda, Some remarks on s-convex functions. Aequationes Math. 48, No. 1, 100—111, (1994).
[38] 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)
[39] U. S. Kirmaci, M. Klarici´c Bakula, M. E Ozdemir and J. Pecari´c, Hadamard-type inequalities for s-convex functions. Appl. Math. Comput. 193, No. 1, pp. 26—35, (2007).
[40] M. A. Latif, On some inequalities for h-convex functions. Int. J. Math. Anal. (Ruse) 4, No. 29-32, pp. 1473—1482, (2010).
[41] G. Maksa and Z. Pales, The equality case in some recent convexity inequalities, Opuscula Math. 31, 2, pp. 269-277, (2011).
[42] D. S. Mitrinovic and I. B. Lackovic, Hermite and convexity, Aequationes Math. 28, pp. 229—232, (1985).
[43] 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, No. 1, pp. 33—36, (1990).
[44] C. E. M. Pearce and A. M. Rubinov, P-functions, quasi-convex functions, and Hadamard-type inequalities. J. Math. Anal. Appl. 240, No. 1, pp. 92—104, (1999).
[45] 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. ”Babe¸s-Bolyai, Cluj-Napoca, (1989).
[46] J. Pecaric and S. S. Dragomir, A generalization of Hadamard’s inequality for isotonic linear functionals, Radovi Mat. (Sarajevo) 7, pp. 103—107, (1991).
[47] M. Radulescu, S. Radulescu and P. Alexandrescu, On the GodunovaLevin-Schur class of functions. Math. Inequal. Appl. 12, No. 4, pp. 853—862, (2009).
[48] M. Z. Sarikaya, A. Saglam, and H. Yildirim, On some Hadamard-type inequalities for h-convex functions. J. Math. Inequal. 2, No. 3, pp. 335—341, (2008).
[49] 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, No. 1, pp. 67—82, (2012).
[50] 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, pp. 265-272, No. 2, (2010).
[51] M. Tun¸c, Ostrowski-type inequalities via h-convex functions with applications to special means. J. Inequal. Appl., 2013, 2013:326. [52] S. Varosanec, On h-convexity. J. Math. Anal. Appl. 326, No. 1, pp. 303-311, (2007).
How to Cite
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S. S. Dragomir, “Inequalities of Hermite-Hadamard Type for h-Convex Functions on Linear Spaces”, Proyecciones (Antofagasta, On line), vol. 34, no. 4, pp. 323-341, 1.
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