Hermite-Hadamard Type Inequalities Related to s-Convex Functions in the Third Sense

Year 2021, Volume: 13 Issue: 2, 318 - 330, 31.12.2021

Zeynep Eken

https://doi.org/10.47000/tjmcs.909498

Cited By: 2

Abstract

In this paper, some Hermite-Hadamard type inequalites for $s$-convex functions in the third sense are studied. It is established several new inequalities for functions whose derivative in absolute value and $p$th power of its derivative in absolute value are $s$-convex in the third sense. In addition, these inequalities are used to find an upper bound for error in numerical integration for this type of functions.

Keywords

Hermite-Hadamard inequality , $s$-convex function in the third sense , convexity

References

• [1] Adilov, G., Yesilce, I., On Generalizations of the Concept of Convexity, Hacettepe Journal of Mathematics and Statistics, 41(2012), 723–730.
• [2] Adilov, G., Yesilce, I., $B^{-1}$-convex Functions, Journal of Convex Analysis, 24(2017), 505–517.
• [3] Adilov, G., Yesilce, I., Some important properties of $B$-convex functions, Journal of Nonlinear and Convex Analysis, 19(2018), 669-680.
• [4] Adilov, G. R., Kemali, S., Abstract convexity and Hermite-Hadamard type inequalities, Journal of Inequalities and Applications, 2009(2009), 943534.
• [5] Alomari, M.W., Darus, M., Kirmaci, U.S., Some inequalities of Hermite-Hadamard type for s-convex functions, Acta Mathematica Scientia, 31(2011), 1643–1652.
• [6] Avriel, M., r-convex Functions, Mathematical Programming, 2(1972), 309–323.
• [7] Bayoumi, A., Fathy Ahmed A., p-Convex Functions in Discrete Sets, International Journal of Engineering and Applied Sciences, 4(2017), 63–66.
• [8] Breckner, W.W., Stetigkeitsaussagen füreine Klasse verallgemekterter konvexer Funktionen in topologischen linearen Raumen, Publ. Inst. Math., 23(1978), 13–20.
• [9] Briec W., Horvath, C., B-convexity, Optimization, 53(2004), 103–127.
• [10] Dragomir, S., Fitzpatrick, S., Hadamard’s inequality for s-convex functions in the first sense and applications, Demonstratio Mathematica, 31(1998), 633–642.
• [11] Dragomir, S. S., Fitzpatrick, S., The Hadamard inequalities for s-convex functions in the second sense, Demonstratio Mathematica, 32(1999), 687–696.
• [12] Eken, Z., Kemali, S., Tinaztepe, G., Adilov G., The Hermite-Hadamard Inequalities for p-Convex Functions, Hacettepe Journal of Mathematics and Statistics, 50(5)(2021), 1268–1279, https://doi.org/10.15672/hujms.775508.
• [13] Gozpinar, A., Set, E., Dragomir, S S., Some generalized Hermite-Hadamard type inequalities involving fractional integral operator for functions whose second derivatives in absolute value are s-convex, Acta Mathematica Universitatis Comenianae, 88(2019), 87–100.
• [14] Kemali, S., Sezer, S., Tinaztepe, G., Adilov, G., s-Convex Functions in the Third Sense, Korean J. Math., 29(3)(2021), 593–602.
• [15] Kemali, S., Tinaztepe, G., Adilov, G., New Type Inequalities For B-Convex Functions Involving Hadamard Fractional Integral, Facta Universitatis-Series Mathematics And Informatics, 33(2018), 697–704.
• [16] Orlicz, W., A note on modular spaces I, Bull. Acad. Polon. Soi., Ser. Math. Astronom Phys., 9(1961), 157-162.
• [17] Özcan, S., Iscan, I., Some new Hermite-Hadamard type inequalities for s-convex functions and their applications, J. Inequal. Appl., 2019(2019), 201.
• [18] Rockafellar, R.T., Convex Analysis, Princeton University Press, New Jersey, 1972.
• [19] Rubinov, A., Abstract Convexity and Global Optimization, Nonconvex Optimization and Its Applications, Kluwer Academic Publisher, Dordrecht, The Netherlands, 2000.
• [20] Sarikaya, M.Z., Kiris, M.E., Some new inequalities of Hermite-Hadamard type for s-convex functions, Miskolc Mathematical Notes, 16(2015), 491–501.
• [21] Set, E., Özdemir, M., Dragomir, S., On Hadamard-type inequalities involving several kinds of convexity, Journal of Inequalities and Applications, (2010), 1–12.
• [22] Set, E., Iscan, I., Kara, H.H., Hermite-Hadamard-Fejer Type Inequalities for s-Convex Function in the Second Sense via Fractional Integrals, Filomat, 30(2016), 3131–3138.
• [23] Sezer, S., Eken, Z., Tinaztepe, G., Adilov, G., p-Convex Functions and Some of Their Properties, Numerical Functional Analysis and Optimization, DOI 10.1080/01630563.2021.1884876, (2021).
• [24] Sezer, S., The Hermite-Hadamard Inequality for s-Convex Functions in the Third Sense, AIMS Mathematics, 6(2021), 7719–7732.
• [25] Sezer, S., Hermite-Hadamard Type Inequalities for the Functions Whose Absolute Values of First Derivatives are p-Convex Function, Fundamental Journal of Mathematics and Applications, 4(2021), 88–99.