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

Yıl 2016, Cilt: 4 Sayı: 1, 54 - 67, 01.04.2016

S. S. Dragomır

Öz

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

Anahtar Kelimeler

Convex functions Integral inequalities h-Convex functions

Kaynakça

  • [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.

Yıl 2016, Cilt: 4 Sayı: 1, 54 - 67, 01.04.2016

S. S. Dragomır

Öz

Kaynakça

  • [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.
Toplam 52 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm Articles
Yazarlar

S. S. Dragomır

Yayımlanma Tarihi 1 Nisan 2016
Gönderilme Tarihi 10 Temmuz 2014
Yayımlandığı Sayı Yıl 2016 Cilt: 4 Sayı: 1

Kaynak Göster

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. Nisan 2016;4(1):54-67.
Chicago Dragomır, S. S. “INEQUALITIES OF HERMITE-HADAMARD TYPE FOR $\phi$-CONVEX FUNCTIONS”. Konuralp Journal of Mathematics 4, sy. 1 (Nisan 2016): 54-67.
EndNote Dragomır SS (01 Nisan 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., c. 4, sy. 1, ss. 54–67, 2016.
ISNAD Dragomır, S. S. “INEQUALITIES OF HERMITE-HADAMARD TYPE FOR $\phi$-CONVEX FUNCTIONS”. Konuralp Journal of Mathematics 4/1 (Nisan 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, c. 4, sy. 1, 2016, ss. 54-67.
Vancouver Dragomır SS. INEQUALITIES OF HERMITE-HADAMARD TYPE FOR $\phi$-CONVEX FUNCTIONS. Konuralp J. Math. 2016;4(1):54-67.
 Kapak Resmi İndir
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