Companions of Hermite-Hadamard Inequality for Convex Functions (II)
DOI:
https://doi.org/10.4067/S0716-09172014000400001Keywords:
Convex functions, Hermite-Hadamard inequality, special means, funciones convexas, desigualdad de Hermite-Hadamard, medias especiales.Abstract
Companions of Hermite-Hadamard inequalities for convex functions defined on the positive axis in the case when the integral has either the weight ψ or 1 ,t > 0 are given. Applications for special means are provided as well.References
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[2] H. ALZER, A note on Hadamard’s inequalities, C.R. Math. Rep. Acad. Sci. Canada, 11, pp. 255-258, (1989).
[3] H. ALZER, On an integral inequality, Math. Rev. Anal. Numer. Theor. Approx., 18, pp. 101-103, (1989).
[4] A. G. AZPEITIA, Convex functions and the Hadamard inequality, Rev.-Colombiana-Mat., 28 (1), pp. 7—12, (1994).
[5] D. BARBU, S. S. DRAGOMIR and C. BUS ¸E, A probabilistic argument for the convergence of some sequences associated to Hadamard’s inequality, Studia Univ. Babe¸ s-Bolyai, Math., 38 (1), pp. 29-33, (1993).
[6] C. BUSE, S. S. DRAGOMIR and D. BARBU, The convergence of some sequences connected to Hadamard’s inequality, Demostratio Math., 29 (1), pp. 53-59, (1996).
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[8] S. S. DRAGOMIR, A refinement of Hadamard’s inequality for isotonic linear functionals, Tamkang J. of Math. (Taiwan), 24 , pp. 101-106. MR 94a: 26043. 2BL No. 799: 26016, (1993).
[9] S. S. DRAGOMIR, On Hadamard’s inequalities for convex functions, Mat. Balkanica, 6, pp. 215-222. MR: 934, (1992). 26033.
[10] S. S. DRAGOMIR, On Hadamard’s inequality for the convex mappings defined on a ball in the space and applications, Math. Ineq. & Appl., 3 (2), pp. 177-187, (2000).
[11] S. S. DRAGOMIR, On Hadamard’s inequality on a disk, Journal of Ineq. in Pure & Appl. Math., 1, No. 1, Article 2, http://jipam.vu.edu.au/, (2000).
[12] S. S. DRAGOMIR, Some integral inequalities for differentiable convex functions, Contributions, Macedonian Acad. of Sci. and Arts, 13 (1), pp. 13-17, (1992).
[13] S. S. DRAGOMIR, Some remarks on Hadamard’s inequalities for convex functions, Extracta Math., 9 (2), pp. 88-94, (1994).
[14] S.S. DRAGOMIR, Two mappings in connection to Hadamard’s inequalities, J. Math. Anal. Appl., 167, pp. 49-56. MR:934:26038, ZBL No. 758:26014, (1992).
[15] S. S. DRAGOMIR, , An inequality improving the first HermiteHadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math. 3, No. 2, Article 31, (2002).
[16] 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, No. 3, Article 35, (2002).
[17] S.S. DRAGOMIR and R. P. AGARWAL, Two new mappings associated with Hadamard’s inequalities for convex functions, Appl. Math. Lett., 11, No. 3, pp. 33-38, (1998).
[18] S. S. DRAGOMIR and C. BUSE, Refinements of Hadamard’s inequality for multiple integrals, Utilitas Math (Canada), 47, pp. 193-195, (1995).
[19] S. S. DRAGOMIR, Y. J. CHO and S. S. KIM, Inequalities of Hadamard’s type for Lipschitzian mappings and their applications, J. of Math. Anal. Appl., 245 (2), pp. 489-501, (2000).
[20] S. S. DRAGOMIR and I. GOMM, Bounds for two mappings associated to the Hermite-Hadamard inequality, Aust. J. Math. Anal. Appl., 8, Art. 5, 9 pages, (2011).
[21] S. S. DRAGOMIR and I. GOMM, Some new bounds for two mappings related to the Hermite-Hadamard inequality for convex functions, Num. Alg. Cont. & Opt. 2, No. 2, pp. 271-278, (2012).
[22] S. S. DRAGOMIR and I. GOMM, Companions of Hermite-Hadamard Inequality for Convex Functions (I), Preprint RGMIA Res. Rep. Coll., 17 (2014), Art.
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[24] S. S. DRAGOMIR and S. FITZPATRICK, The Hadamard’s inequality for s-convex functions in the second sense, Demonstratio Math., 32 (4), pp. 687-696, (1999).
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[26] S. S. DRAGOMIR, D. S. MILOSEVIC and J. SANDOR, On some refinements of Hadamard’s inequalities and applications, Univ. Belgrad, Publ. Elek. Fak. Sci. Math., 4, pp. 21-24, (1993).
[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, Quasi-convex functions and Hadamard’s inequality, Bull. Austral. Math. Soc., 57 , pp. 377-385, (1998).
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[30] S. S. DRAGOMIR, C. E. M. PEARCE and J. E. PECARIC, On Jessen’s and related inequalities for isotonic sublinear functionals, Acta Math. Sci. (Szeged), 61, pp. 373-382, (1995).
[31] S. S. DRAGOMIR, J. E. PECARIC and L. E. PERSSON, Some inequalities of Hadamard type, Soochow J. of Math. (Taiwan), 21, pp. 335-341, (1995).
[32] S. S. DRAGOMIR, J. E. PECARIC and J. SANDOR, A note on the Jensen-Hadamard inequality, Anal. Num. Theor. Approx., 19, pp. 21-28. MR 93b : 260 14.ZBL No. 733 : 26010, (1990).
[33] S. S. DRAGOMIR and G. H. TOADER, Some inequalities for m-convex functions, Studia Univ. Babe¸ s-Bolyai, Math., 38 (1), pp. 21-28, (1993).
[34] A. M. FINK, A best possible Hadamard inequality, Math. Ineq. & Appl., 2 , pp. 223-230, (1998).
[35] A. M. FINK, Toward a theory of best possible inequalities, Nieuw Archief von Wiskunde, 12 , pp. 19-29, (1994).
[36] A. M. FINK, Two inequalities, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat., 6, pp. 48-49, (1995).
[37] B. GAVREA, On Hadamard’s inequality for the convex mappings defined on a convex domain in the space, Journal of Ineq. in Pure & Appl. Math., 1 (2000), No. 1, Article 9, http://jipam.vu.edu.au/
[38] P. M. GILL, C. E. M. PEARCE and J. PECARIC, Hadamard’s inequality for r-convex functions, J. of Math. Anal. and Appl., 215 , pp. 461-470, (1997).
[39] G. H. HARDY, J. E. LITTLEWOOD and G. POLYA, Inequalities, 2nd Ed., Cambridge University Press, (1952).
[40] K.-C. LEE and K.-L. TSENG, On a weighted generalisation of Hadamard’s inequality for G-convex functions, Tamsui Oxford Journal of Math. Sci., 16 (1), pp. 91-104, (2000).
[41] A. LUPAS, The Jensen-Hadamard inequality for convex functions of higher order, Octogon Math. Mag., 5, No. 2, pp. 8-9, (1997).
[42] A. LUPAS, A generalisation of Hadamard’s inequality for convex functions, Univ. Beograd. Publ. Elek. Fak. Ser. Mat. Fiz., No. 544-576, pp. 115-121, (1976).
[43] D. M. MAKISIMOVIC, A short proof of generalized Hadamard’s inequalities, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 634-677, pp. 126—128, (1979).
[44] D.S. MITRINOVIC and I. LACKOVIC, Hermite and convexity, Aequat. Math., 28, pp. 229—232, (1985).
[45] D. S. MITRINOVIC, J. E. PECARIC and A. M. FINK, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht/Boston/London.
[46] E. NEUMAN, Inequalities involving generalised symmetric means, J. Math. Anal. Appl., 120, pp. 315-320, (1986).
[47] E. NEUMAN and J. E. PECARIC, Inequalities involving multivariate convex functions, J. Math. Anal. Appl., 137, pp. 514-549, (1989).
[48] E. NEUMAN, Inequalities involving multivariate convex functions II, Proc. Amer. Math. Soc., 109, pp. 965-974, (1990).
[49] C. P. NICULESCU, A note on the dual Hermite-Hadamard inequality, The Math. Gazette, July (2000).
[50] C. P. NICULESCU, Convexity according to the geometric mean, Math. Ineq. & Appl., 3 (2), pp. 155-167, (2000).
[51] C. E. M. PEARCE, J. PECARIC and V. SIMIC, Stolarsky means and Hadamard’s inequality, J. Math. Anal. Appl., 220, pp. 99-109, (1998).
[52] C. E. M. PEARCE and A. M. RUBINOV, P-functions, quasi-convex functions and Hadamard-type inequalities, J. Math. Anal. Appl., 240, (1), pp. 92-104, (1999).
[53] J. E. PECARIC, Remarks on two interpolations of Hadamard’s inequalities, Contributions, Macedonian Acad. of Sci. and Arts, Sect. of Math. and Technical Sciences, (Scopje), 13, pp. 9-12, (1992).
[54] J. PECARIC and S. S. DRAGOMIR, A generalization of Hadamard’s integral inequality for isotonic linear functionals, Rudovi Mat. (Sarajevo), 7 (1991), 103-107. MR 924: 26026. 2BL No. 738: 26006.
[55] J. PECARIC, F. PROSCHAN and Y. L. TONG, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Inc., (1992).
[56] J. SANDOR, A note on the Jensen-Hadamard inequality, Anal. Numer. Theor. Approx., 19, No. 1, pp. 29-34, (1990).
[57] J. SANDOR, An application of the Jensen-Hadamard inequality, Nieuw-Arch.-Wisk., 8, No. 1, pp. 63-66, (1990).
[58] J. SANDOR, On the Jensen-Hadamard inequality, Studia Univ. BabesBolyai, Math., 36, No. 1, pp. 9-15, (1991).
[59] P. M. VASIC, I. B. LACKOVIC and D. M. MAKSIMOVIC, Note on convex functions IV: OnHadamard’s inequality for weighted arithmetic means, Univ. Beograd Publ. Elek. Fak., Ser. Mat. Fiz., No. 678-715, pp. 199-205, (1980).
[60] G. S. YANG and M. C. HONG, A note on Hadamard’s inequality, Tamkang J. Math., 28 (1), pp. 33-37, (1997).
[61] G. S. YANG and K. L. TSENG, On certain integral inequalities related to Hermite-Hadamard inequalities, J. Math. Anal. Appl., 239, pp. 180-187, (1999).
[2] H. ALZER, A note on Hadamard’s inequalities, C.R. Math. Rep. Acad. Sci. Canada, 11, pp. 255-258, (1989).
[3] H. ALZER, On an integral inequality, Math. Rev. Anal. Numer. Theor. Approx., 18, pp. 101-103, (1989).
[4] A. G. AZPEITIA, Convex functions and the Hadamard inequality, Rev.-Colombiana-Mat., 28 (1), pp. 7—12, (1994).
[5] D. BARBU, S. S. DRAGOMIR and C. BUS ¸E, A probabilistic argument for the convergence of some sequences associated to Hadamard’s inequality, Studia Univ. Babe¸ s-Bolyai, Math., 38 (1), pp. 29-33, (1993).
[6] C. BUSE, S. S. DRAGOMIR and D. BARBU, The convergence of some sequences connected to Hadamard’s inequality, Demostratio Math., 29 (1), pp. 53-59, (1996).
[7] S. S. DRAGOMIR, A mapping in connection to Hadamard’s inequalities, An. Oster. Akad. Wiss. Math.-Natur., (Wien), 128, pp. 17-20. MR 934:26032. ZBL No. 747:26015, (1991).
[8] S. S. DRAGOMIR, A refinement of Hadamard’s inequality for isotonic linear functionals, Tamkang J. of Math. (Taiwan), 24 , pp. 101-106. MR 94a: 26043. 2BL No. 799: 26016, (1993).
[9] S. S. DRAGOMIR, On Hadamard’s inequalities for convex functions, Mat. Balkanica, 6, pp. 215-222. MR: 934, (1992). 26033.
[10] S. S. DRAGOMIR, On Hadamard’s inequality for the convex mappings defined on a ball in the space and applications, Math. Ineq. & Appl., 3 (2), pp. 177-187, (2000).
[11] S. S. DRAGOMIR, On Hadamard’s inequality on a disk, Journal of Ineq. in Pure & Appl. Math., 1, No. 1, Article 2, http://jipam.vu.edu.au/, (2000).
[12] S. S. DRAGOMIR, Some integral inequalities for differentiable convex functions, Contributions, Macedonian Acad. of Sci. and Arts, 13 (1), pp. 13-17, (1992).
[13] S. S. DRAGOMIR, Some remarks on Hadamard’s inequalities for convex functions, Extracta Math., 9 (2), pp. 88-94, (1994).
[14] S.S. DRAGOMIR, Two mappings in connection to Hadamard’s inequalities, J. Math. Anal. Appl., 167, pp. 49-56. MR:934:26038, ZBL No. 758:26014, (1992).
[15] S. S. DRAGOMIR, , An inequality improving the first HermiteHadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math. 3, No. 2, Article 31, (2002).
[16] 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, No. 3, Article 35, (2002).
[17] S.S. DRAGOMIR and R. P. AGARWAL, Two new mappings associated with Hadamard’s inequalities for convex functions, Appl. Math. Lett., 11, No. 3, pp. 33-38, (1998).
[18] S. S. DRAGOMIR and C. BUSE, Refinements of Hadamard’s inequality for multiple integrals, Utilitas Math (Canada), 47, pp. 193-195, (1995).
[19] S. S. DRAGOMIR, Y. J. CHO and S. S. KIM, Inequalities of Hadamard’s type for Lipschitzian mappings and their applications, J. of Math. Anal. Appl., 245 (2), pp. 489-501, (2000).
[20] S. S. DRAGOMIR and I. GOMM, Bounds for two mappings associated to the Hermite-Hadamard inequality, Aust. J. Math. Anal. Appl., 8, Art. 5, 9 pages, (2011).
[21] S. S. DRAGOMIR and I. GOMM, Some new bounds for two mappings related to the Hermite-Hadamard inequality for convex functions, Num. Alg. Cont. & Opt. 2, No. 2, pp. 271-278, (2012).
[22] S. S. DRAGOMIR and I. GOMM, Companions of Hermite-Hadamard Inequality for Convex Functions (I), Preprint RGMIA Res. Rep. Coll., 17 (2014), Art.
[23] S. S. DRAGOMIR and S. FITZPATRICK, The Hadamard’s inequality for s-convex functions in the first sense, Demonstratio Math., 31 (3), pp. 633-642, (1998).
[24] S. S. DRAGOMIR and S. FITZPATRICK, The Hadamard’s inequality for s-convex functions in the second sense, Demonstratio Math., 32 (4), pp. 687-696, (1999).
[25] S. S. DRAGOMIR and N. M. IONESCU, On some inequalities for convex-dominated functions, Anal. Num. Theor. Approx., 19, pp. 21-28. MR 936: 26014 ZBL No. 733 : 26010, (1990).
[26] S. S. DRAGOMIR, D. S. MILOSEVIC and J. SANDOR, On some refinements of Hadamard’s inequalities and applications, Univ. Belgrad, Publ. Elek. Fak. Sci. Math., 4, pp. 21-24, (1993).
[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, Quasi-convex functions and Hadamard’s inequality, Bull. Austral. Math. Soc., 57 , pp. 377-385, (1998).
[29] S. S. DRAGOMIR and C. E. M. PEARCE, Selected Topics on HermiteHadamard Inequalities and Applications, RGMIA Monographs, (2000). [Online http://rgmia.org/monographs/hermite hadamard.html].
[30] S. S. DRAGOMIR, C. E. M. PEARCE and J. E. PECARIC, On Jessen’s and related inequalities for isotonic sublinear functionals, Acta Math. Sci. (Szeged), 61, pp. 373-382, (1995).
[31] S. S. DRAGOMIR, J. E. PECARIC and L. E. PERSSON, Some inequalities of Hadamard type, Soochow J. of Math. (Taiwan), 21, pp. 335-341, (1995).
[32] S. S. DRAGOMIR, J. E. PECARIC and J. SANDOR, A note on the Jensen-Hadamard inequality, Anal. Num. Theor. Approx., 19, pp. 21-28. MR 93b : 260 14.ZBL No. 733 : 26010, (1990).
[33] S. S. DRAGOMIR and G. H. TOADER, Some inequalities for m-convex functions, Studia Univ. Babe¸ s-Bolyai, Math., 38 (1), pp. 21-28, (1993).
[34] A. M. FINK, A best possible Hadamard inequality, Math. Ineq. & Appl., 2 , pp. 223-230, (1998).
[35] A. M. FINK, Toward a theory of best possible inequalities, Nieuw Archief von Wiskunde, 12 , pp. 19-29, (1994).
[36] A. M. FINK, Two inequalities, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat., 6, pp. 48-49, (1995).
[37] B. GAVREA, On Hadamard’s inequality for the convex mappings defined on a convex domain in the space, Journal of Ineq. in Pure & Appl. Math., 1 (2000), No. 1, Article 9, http://jipam.vu.edu.au/
[38] P. M. GILL, C. E. M. PEARCE and J. PECARIC, Hadamard’s inequality for r-convex functions, J. of Math. Anal. and Appl., 215 , pp. 461-470, (1997).
[39] G. H. HARDY, J. E. LITTLEWOOD and G. POLYA, Inequalities, 2nd Ed., Cambridge University Press, (1952).
[40] K.-C. LEE and K.-L. TSENG, On a weighted generalisation of Hadamard’s inequality for G-convex functions, Tamsui Oxford Journal of Math. Sci., 16 (1), pp. 91-104, (2000).
[41] A. LUPAS, The Jensen-Hadamard inequality for convex functions of higher order, Octogon Math. Mag., 5, No. 2, pp. 8-9, (1997).
[42] A. LUPAS, A generalisation of Hadamard’s inequality for convex functions, Univ. Beograd. Publ. Elek. Fak. Ser. Mat. Fiz., No. 544-576, pp. 115-121, (1976).
[43] D. M. MAKISIMOVIC, A short proof of generalized Hadamard’s inequalities, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 634-677, pp. 126—128, (1979).
[44] D.S. MITRINOVIC and I. LACKOVIC, Hermite and convexity, Aequat. Math., 28, pp. 229—232, (1985).
[45] D. S. MITRINOVIC, J. E. PECARIC and A. M. FINK, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht/Boston/London.
[46] E. NEUMAN, Inequalities involving generalised symmetric means, J. Math. Anal. Appl., 120, pp. 315-320, (1986).
[47] E. NEUMAN and J. E. PECARIC, Inequalities involving multivariate convex functions, J. Math. Anal. Appl., 137, pp. 514-549, (1989).
[48] E. NEUMAN, Inequalities involving multivariate convex functions II, Proc. Amer. Math. Soc., 109, pp. 965-974, (1990).
[49] C. P. NICULESCU, A note on the dual Hermite-Hadamard inequality, The Math. Gazette, July (2000).
[50] C. P. NICULESCU, Convexity according to the geometric mean, Math. Ineq. & Appl., 3 (2), pp. 155-167, (2000).
[51] C. E. M. PEARCE, J. PECARIC and V. SIMIC, Stolarsky means and Hadamard’s inequality, J. Math. Anal. Appl., 220, pp. 99-109, (1998).
[52] C. E. M. PEARCE and A. M. RUBINOV, P-functions, quasi-convex functions and Hadamard-type inequalities, J. Math. Anal. Appl., 240, (1), pp. 92-104, (1999).
[53] J. E. PECARIC, Remarks on two interpolations of Hadamard’s inequalities, Contributions, Macedonian Acad. of Sci. and Arts, Sect. of Math. and Technical Sciences, (Scopje), 13, pp. 9-12, (1992).
[54] J. PECARIC and S. S. DRAGOMIR, A generalization of Hadamard’s integral inequality for isotonic linear functionals, Rudovi Mat. (Sarajevo), 7 (1991), 103-107. MR 924: 26026. 2BL No. 738: 26006.
[55] J. PECARIC, F. PROSCHAN and Y. L. TONG, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Inc., (1992).
[56] J. SANDOR, A note on the Jensen-Hadamard inequality, Anal. Numer. Theor. Approx., 19, No. 1, pp. 29-34, (1990).
[57] J. SANDOR, An application of the Jensen-Hadamard inequality, Nieuw-Arch.-Wisk., 8, No. 1, pp. 63-66, (1990).
[58] J. SANDOR, On the Jensen-Hadamard inequality, Studia Univ. BabesBolyai, Math., 36, No. 1, pp. 9-15, (1991).
[59] P. M. VASIC, I. B. LACKOVIC and D. M. MAKSIMOVIC, Note on convex functions IV: OnHadamard’s inequality for weighted arithmetic means, Univ. Beograd Publ. Elek. Fak., Ser. Mat. Fiz., No. 678-715, pp. 199-205, (1980).
[60] G. S. YANG and M. C. HONG, A note on Hadamard’s inequality, Tamkang J. Math., 28 (1), pp. 33-37, (1997).
[61] G. S. YANG and K. L. TSENG, On certain integral inequalities related to Hermite-Hadamard inequalities, J. Math. Anal. Appl., 239, pp. 180-187, (1999).
Published
2017-03-23
How to Cite
[1]
S. S. Dragomir and I. Gomm, “Companions of Hermite-Hadamard Inequality for Convex Functions (II)”, Proyecciones (Antofagasta, On line), vol. 33, no. 4, pp. 349-367, Mar. 2017.
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