HADAMARD, SIMPSON AND OSTROWSKI TYPE INEQUALITIES FOR E-CONVEXITY

Year 2019, Volume: 3 Issue: 2, 141 - 158, 30.04.2019

Musa Çakmak

https://doi.org/10.26900/jsp.3.015

Abstract

In this study, we proposed a new definition to give a
different perspec-tive to convex functions. We have introduced the expansion of
Hadamard, midpoint Hadamard, trapezoid Hadamard, Simpson and Ostrowski
inequalities for the newly defined classes of convex functions.

Keywords

(a; m; e)-convexity , Hadamard's inequality , Simpson inequality , Ostrowski inequality.

References

• [1] ALOMARİ M., DARUS M., KIRMACIU.S., Refinements of Hadamard-type inequalities for quasi-convex functions with applications to trapezoidal formula and to special means, Comp. and Math. with Appl., V.59, 2010, pp. 225-232.
• [2] ALOMARİ M., DARUS M., DRAGOMİR S.S., New inequalities of Simpson's type for s-convex functions with applications, RGMIA Research Report Collection Volume 12 (4), 2010.
• [3] ALOMARİ M., DARUS M., Some Ostrowski type inequalities for quasi-convex functions with applications to special means, RGMIA Research Report Collection Volume 13, article 2, 2010.
• [4] BESSENYEİ M., The Hermite-Hadamard inequality on simplices, Amer. Math. Monthly 115 (2008), no. 4, 339-345. MR 2009b:52023
• [5] BESSENYEİ M., Hermite-Hadamard-type inequalities for generalized convex functions, J. Inequal. Pure Appl. Math. 9 (2008), no. 3, Article 63, pp. 51 (electronic).
• [6] BESSENYEİ M., The Hermite-Hadamard inequality in Beckenbach's setting, J. Math. Anal. Appl. 364 (2010), no. 2, 366-383. MR MR2576189
• [7] BESSENYEİ M. and PÁLES Zs., Higher-order generalizations of Hadamard's inequality, Publ. Math. Debrecen 61 (2002), no. 3-4, 623-643. MR 2003k:26021
• [8] BESSENYEİ M. and PÁLES Zs., Characterizations of convexity via Hadamard's inequality, Math. Inequal. Appl. 9 (2006), no. 1, 53-62. MR 2007a:26010
• [9] BOMBARDELLİ M. and VAROŠANEC S., Properties of h -convex functions related to the Hermite-Hadamard-Fejér inequalities, Comput. Math. Appl. 58 (2009) 1869-1877.
• [10] DRAGOMİR S. S., AGARWAL R. P., and CERONE P., On Simpson's inequality and applications, J. Inequal. Appl. 5 (2000), no. 6, 533{579; Available online at http://dx.doi.org/10.1155/S102558340000031X.
• [11] DRAGOMİR S. S., PEČARİĆ J. and PERSSON L.E., Some inequalities of Hadamard type, Soochow J.Math., 21, 335-241, 1995.
• [12] GÖZPİNAR A., SET E., DRAGOMİR S. S., some generalized Hermite-Hadamard type inequalities involving fractional integral operator for functions whose second derivatives in absolute value are s-convex, Acta Math. Univ. Comenianae, in press
• [13] HUDZİK H. and MALİGRANDA L., Some remarks on s-convex functions, Aequationes Math., 48, 100-111, 1994.
• [14] KIRMACI U.S., Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula, Appl.Math.Comput. 147 (1), 137-146 (2004).
• [15] MAKÓ J. and PÁLES Zs., Approximate Hermite-Hadamard type inequalities for approximately convex functions, Math. Inequal. Appl., 16 (2), 507-526, (2013).
• [16] MAKÓ J. and PÁLES Zs., Implications between approximate convexity properties and approximate Hermite-Hadamard inequalities, Cent. Eur. J. Math. 10 (2012), no. 3, 1017-1041.
• [17] MİTRİNOVİĆ D. S., PEČARİĆ J. , and FİNK A.M., Classical and new inequalities in analysis, KluwerAcademic, Dordrecht, 1993.
• [18] ÖZDEMİR M.E., GÜRBÜZ M. and AKDEMİR A.O., Inequalities for h-Convex Functions via Further Properties, RGMIA Research Report Collection Volume 14, article 22, 2011.
• [19] SARIKAYA M.Z., SAĞLAM A. and YILDIRIM H., some Hadamard-type inequalities for h-convex functions, J. Math. Inequal. 2 (3) (2008) 335-341.
• [20] NOOR M.A., NOOR K.I., IFTİKHAR S. On Integral Inequalities of Hermite-Hadamard Type for Harmonic (h,s)-Convex Functions, International Journal of Analysis and Applications 11(1) (2016), 61-69.
• [21] SARIKAYA M.Z., SET E. and ÖZDEMİR M.E., On some new inequalities of Hadamard-type involving h-convex functions, Acta Math. Univ. Comenian LXXIX (2) (2010) 265-272.
• [22] PACHPATTE B. G., Mathematical Inequalities, North-Holland Mathematical Library, Elsevier Science B.V. Amsterdam, 2005.
• [23] PEČARİĆ J.E., PROSCHAN F., TONG Y.L., Convex Functions, Partial Orderings and Statistical Applications, Academic Press, 1991.
• [24] TOADER G.H., ON a generalisation of the convexity, Mathematica, 30 (53) (1988), 83-87.
• [25] TOADER G.H., Some generalisations of the convexity, Proc. Colloq. Approx. Optim, Cluj-Napoca (Romania), 1984, 329-338.
• [26] TUNÇ M. , Ostrowski-type inequalities via h-convex functions with applications to special means, Jour. Ineq. and Appl. 2013 (1), 326, 2013.
• [27] VAROŠANEC S., On h-convexity, J. Math. Anal. Appl., Volume 326, Issue 1 (2007), 303-311.