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INEQUALITIES OF HERMITE-HADAMARD TYPE FOR $\phi$-CONVEX FUNCTIONS

Year 2016, Volume: 4 Issue: 1, 54 - 67, 01.04.2016

S. S. Dragomır

Abstract

Some inequalities of Hermite-Hadamard type for '-convex functions de ned on real intervals are given.

Keywords

Convex functions Integral inequalities h-Convex functions

References

  • [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. 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].
  • [5] E. F. Beckenbach, Convex functions, Bull. Amer. Math. Soc. 54(1948), 439{460.
  • [6] 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.
  • [7] 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.
  • [8] 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.
  • [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, 2002, 53-62.
  • [11] P. Cerone, S. S. Dragomir and J. Roumeliotis, Some Ostrowski type inequalities for n-time di erentiable mappings and applications, Demonstratio Mathematica, 32(2) (1999), 697| 712.
  • [12] G. Cristescu, Hadamard type inequalities for convolution of h-convex functions. Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity 8 (2010), 3{11.
  • [13] S. S. Dragomir, Ostrowski's inequality for monotonous mappings and applications, J. KSIAM, 3(1) (1999), 127-135.
  • [14] S. S. Dragomir, The Ostrowski's integral inequality for Lipschitzian mappings and applications, Comp. Math. Appl., 38 (1999), 33-37.
  • [15] S. S. Dragomir, On the Ostrowski's inequality for Riemann-Stieltjes integral, Korean J. Appl. Math., 7 (2000), 477-485.
  • [16] S. S. Dragomir, On the Ostrowski's inequality for mappings of bounded variation and applications, Math. Ineq. & Appl., 4(1) (2001), 33-40.
  • [17] S. S. Dragomir, On the Ostrowski inequality for Riemann-Stieltjes integral R b a f (t) du (t) where f is of Holder type and u is of bounded variation and applications, J. KSIAM, 5(1) (2001), 35-45.
  • [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 rst Hermite-Hadamard inequality for convex functions de ned 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 rst Hermite-Hadamard inequality for convex functions de ned 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 Hermite-Hadamard inequality for convex functions de ned 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) (2003), 373-382.
  • [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, Inequalities of Hermite-Hadamard type for h-convex functions on linear spaces, Preprint RGMIA Res. Rep. Coll. 16 (2013), Art. 72 [Online http://rgmia.org/papers/v16/v16a72.pdf].
  • [25] 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.
  • [26] 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.
  • [27] 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.
  • [28] 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.
  • [29] 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.
  • [30] S. S. Dragomir and C. E. M. Pearce, Quasi-convex functions and Hadamard's inequality, Bull. Austral. Math. Soc. 57 (1998), 377-385.
  • [31] S. S. Dragomir, J. Pecaric and L. Persson, Some inequalities of Hadamard type. Soochow J. Math. 21 (1995), no. 3, 335{341.
  • [32] S. S. Dragomir and Th. M. Rassias (Eds), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publisher, 2002.
  • [33] 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.
  • [34] 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.
  • [35] 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.
  • [36] A. El Farissi, Simple proof and refeinment of Hermite-Hadamard inequality, J. Math. Ineq. 4 (2010), No. 3, 365{369.
  • [37] 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
  • [38] H. Hudzik and L. Maligranda, Some remarks on s-convex functions. Aequationes Math. 48 (1994), no. 1, 100{111.
  • [39] 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).
  • [40] 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.
  • [41] M. A. Latif, On some inequalities for h-convex functions. Int. J. Math. Anal. (Ruse) 4 (2010), no. 29-32, 1473{1482.
  • [42] D. S. Mitrinovic and I. B. Lackovic, Hermite and convexity, Aequationes Math. 28 (1985), 229{232.
  • [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 (1990), no. 1, 33{36.
  • [44] 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.
  • [45] J. E. Pecaric and S. S. Dragomir, On an inequality of Godunova-Levin and some re nements 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.
  • [46] J. Pecaric and S. S. Dragomir, A generalization of Hadamard's inequality for isotonic linear functionals, Radovi Mat. (Sarajevo) 7 (1991), 103{107.
  • [47] M. Radulescu, S. Radulescu and P. Alexandrescu, On the Godunova-Levin-Schur class of functions. Math. Inequal. Appl. 12 (2009), no. 4, 853{862.
  • [48] M. Z. Sarikaya, A. Saglam, and H. Yildirim, On some Hadamard-type inequalities for hconvex functions. J. Math. Inequal. 2 (2008), no. 3, 335{341.
  • [49] E. Set, M. E.  Ozdemir and M. Z. Sarkaya, 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.
  • [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 (2010), no. 2, 265{272.
  • [51] M. Tunc, 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 (2007), no. 1, 303{311.

Year 2016, Volume: 4 Issue: 1, 54 - 67, 01.04.2016

S. S. Dragomır

Abstract

References

  • [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. 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].
  • [5] E. F. Beckenbach, Convex functions, Bull. Amer. Math. Soc. 54(1948), 439{460.
  • [6] 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.
  • [7] 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.
  • [8] 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.
  • [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, 2002, 53-62.
  • [11] P. Cerone, S. S. Dragomir and J. Roumeliotis, Some Ostrowski type inequalities for n-time di erentiable mappings and applications, Demonstratio Mathematica, 32(2) (1999), 697| 712.
  • [12] G. Cristescu, Hadamard type inequalities for convolution of h-convex functions. Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity 8 (2010), 3{11.
  • [13] S. S. Dragomir, Ostrowski's inequality for monotonous mappings and applications, J. KSIAM, 3(1) (1999), 127-135.
  • [14] S. S. Dragomir, The Ostrowski's integral inequality for Lipschitzian mappings and applications, Comp. Math. Appl., 38 (1999), 33-37.
  • [15] S. S. Dragomir, On the Ostrowski's inequality for Riemann-Stieltjes integral, Korean J. Appl. Math., 7 (2000), 477-485.
  • [16] S. S. Dragomir, On the Ostrowski's inequality for mappings of bounded variation and applications, Math. Ineq. & Appl., 4(1) (2001), 33-40.
  • [17] S. S. Dragomir, On the Ostrowski inequality for Riemann-Stieltjes integral R b a f (t) du (t) where f is of Holder type and u is of bounded variation and applications, J. KSIAM, 5(1) (2001), 35-45.
  • [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 rst Hermite-Hadamard inequality for convex functions de ned 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 rst Hermite-Hadamard inequality for convex functions de ned 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 Hermite-Hadamard inequality for convex functions de ned 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) (2003), 373-382.
  • [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, Inequalities of Hermite-Hadamard type for h-convex functions on linear spaces, Preprint RGMIA Res. Rep. Coll. 16 (2013), Art. 72 [Online http://rgmia.org/papers/v16/v16a72.pdf].
  • [25] 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.
  • [26] 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.
  • [27] 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.
  • [28] 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.
  • [29] 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.
  • [30] S. S. Dragomir and C. E. M. Pearce, Quasi-convex functions and Hadamard's inequality, Bull. Austral. Math. Soc. 57 (1998), 377-385.
  • [31] S. S. Dragomir, J. Pecaric and L. Persson, Some inequalities of Hadamard type. Soochow J. Math. 21 (1995), no. 3, 335{341.
  • [32] S. S. Dragomir and Th. M. Rassias (Eds), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publisher, 2002.
  • [33] 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.
  • [34] 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.
  • [35] 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.
  • [36] A. El Farissi, Simple proof and refeinment of Hermite-Hadamard inequality, J. Math. Ineq. 4 (2010), No. 3, 365{369.
  • [37] 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
  • [38] H. Hudzik and L. Maligranda, Some remarks on s-convex functions. Aequationes Math. 48 (1994), no. 1, 100{111.
  • [39] 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).
  • [40] 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.
  • [41] M. A. Latif, On some inequalities for h-convex functions. Int. J. Math. Anal. (Ruse) 4 (2010), no. 29-32, 1473{1482.
  • [42] D. S. Mitrinovic and I. B. Lackovic, Hermite and convexity, Aequationes Math. 28 (1985), 229{232.
  • [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 (1990), no. 1, 33{36.
  • [44] 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.
  • [45] J. E. Pecaric and S. S. Dragomir, On an inequality of Godunova-Levin and some re nements 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.
  • [46] J. Pecaric and S. S. Dragomir, A generalization of Hadamard's inequality for isotonic linear functionals, Radovi Mat. (Sarajevo) 7 (1991), 103{107.
  • [47] M. Radulescu, S. Radulescu and P. Alexandrescu, On the Godunova-Levin-Schur class of functions. Math. Inequal. Appl. 12 (2009), no. 4, 853{862.
  • [48] M. Z. Sarikaya, A. Saglam, and H. Yildirim, On some Hadamard-type inequalities for hconvex functions. J. Math. Inequal. 2 (2008), no. 3, 335{341.
  • [49] E. Set, M. E.  Ozdemir and M. Z. Sarkaya, 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.
  • [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 (2010), no. 2, 265{272.
  • [51] M. Tunc, 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 (2007), no. 1, 303{311.
There are 52 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

S. S. Dragomır

Publication Date April 1, 2016
Submission Date July 10, 2014
Published in Issue Year 2016 Volume: 4 Issue: 1

Cite

APA Dragomır, S. S. (2016). INEQUALITIES OF HERMITE-HADAMARD TYPE FOR $\phi$-CONVEX FUNCTIONS. Konuralp Journal of Mathematics, 4(1), 54-67.
AMA Dragomır SS. INEQUALITIES OF HERMITE-HADAMARD TYPE FOR $\phi$-CONVEX FUNCTIONS. Konuralp J. Math. April 2016;4(1):54-67.
Chicago Dragomır, S. S. “INEQUALITIES OF HERMITE-HADAMARD TYPE FOR $\phi$-CONVEX FUNCTIONS”. Konuralp Journal of Mathematics 4, no. 1 (April 2016): 54-67.
EndNote Dragomır SS (April 1, 2016) INEQUALITIES OF HERMITE-HADAMARD TYPE FOR $\phi$-CONVEX FUNCTIONS. Konuralp Journal of Mathematics 4 1 54–67.
IEEE S. S. Dragomır, “INEQUALITIES OF HERMITE-HADAMARD TYPE FOR $\phi$-CONVEX FUNCTIONS”, Konuralp J. Math., vol. 4, no. 1, pp. 54–67, 2016.
ISNAD Dragomır, S. S. “INEQUALITIES OF HERMITE-HADAMARD TYPE FOR $\phi$-CONVEX FUNCTIONS”. Konuralp Journal of Mathematics 4/1 (April 2016), 54-67.
JAMA Dragomır SS. INEQUALITIES OF HERMITE-HADAMARD TYPE FOR $\phi$-CONVEX FUNCTIONS. Konuralp J. Math. 2016;4:54–67.
MLA Dragomır, S. S. “INEQUALITIES OF HERMITE-HADAMARD TYPE FOR $\phi$-CONVEX FUNCTIONS”. Konuralp Journal of Mathematics, vol. 4, no. 1, 2016, pp. 54-67.
Vancouver Dragomır SS. INEQUALITIES OF HERMITE-HADAMARD TYPE FOR $\phi$-CONVEX FUNCTIONS. Konuralp J. Math. 2016;4(1):54-67.
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