Hermite-Hadamard Type Inequality for s-Convex Functions in the Fourth Sense

Abstract

In this study, firstly, Hermite-Hadamard type inequalities are examined for functions whose first derivative is $s$-convex functions in the fourth sense. In addition, Hermite-Hadamard type inequalities are examined for functions whose second derivative is $s$-convex functions in the fourth sense. Finally, some application examples including special tools and digamma functions are presented.

Keywords

• Convex functions
• s-convexity
• Hermite-Hadamard type inequality
• specials means
• Digamma functions

References

1. [1] Abramowitz, M., Stegun, I.A., (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, Washington, 1970.
2. [2] Adilov, G., Rubinov, A.M., $B-$convex sets and functions. Numerical Functional Analysis and Optimization, 27(3-4)(2006), 237–257.
3. [3] Adilov, G., Yesilce, I., On Generalizations of the concept of convexity, Hacettepe Journal of Mathematics and Statistics, 41(5)(2012), 723–730.
4. [4] Adilov, G., Kemali, S., Abstract convexity and Hermite-Hadamard type inequalities, Journal of Inequalities and Applications, 2009(2009), 943534.
5. [5] Alomari, M.W., Darus, M., Kirmaci, U.S., Some inequalities of Hermite-Hadamard type for s-convex functions, Acta Mathematica Scientia, 31(4)(2011), 1643–1652.
6. [6] Dragomir, S.S., Agarwal, R.P.,Two Inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11(5)(1998), 91–95.
7. [7] Dragomir, S.S., Pierce, C.E.M., On some inequalities for differentiable convex functions and applications, preprint, (2000).
8. [8] Dragomir, S.S., Pierce, C.E.M., Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA, Victoria University, 2000.

9. [9] Dragomir S.S., Fitzpatrick, S., The Hadamard’s inequality for s-convex functions in the second sense, Demonstratio Math., 32(4)(1999), 687–696.
10. [10] Eken, Z., Sezer, S., Tınaztepe, G., Adilov, G., s-convex functions in the fourth sense and their some properties. Submitted.
11. [11] Hudzik, H., Maligranda, L., Some remarks on s-convex functions. Aequ. Math. 48(1994), 100–111.
12. [12] Khan, M., Hanif, M.A., Khan, A.H., et al, Association of Jensen’s inequality for s-convex function with Csiszar divergence, J Inequal 162(2019).
13. [13] Kırmacı, U.S., Bakula, M.K., Ozdemir, M.E., Pecaric, J., Hadamard-type inequalities for s-convex functions, Appl. Math. Comput., 193(2007), 26–35.
14. [14] Kemali, S., Sezer, S., Tınaztepe, G., Adilov, G., s-Convex function in the third sense, Korean J. Math., 29(3)(2021), 593-602.
15. [15] Kemali, S., Yesilce, I., Adilov, G., $B$-Convexity, $B^{-1}$-Convexity, and their comparison, Numerical Functional Analysis and Optimization, 36(2)(2015), 133–146.
16. [16] Orlicz, W., A note on modular spaces I. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys, 9(1961), 157–162.
17. [17] Özcan, S., İşcan, I., Some new Hermite–Hadamard type inequalities for s-convex functions and their applications, Journal of Inequalities and Applications, Article, 201, (2019).
18. [18] Pearce C.E.M., Pecaric, J., Inequalities for differentiable mappings with application to special means and quadrature formula, Applied Mathematics Letters 13(2)(2000), 51–55.
19. [19] Sezer, S., Eken, Z., Tınaztepe, G., Adilov, G., p-Convex functions and some of their properties, Numerical Functional Analysis and Optimization, 42(4)(2021), 443–459.
20. [20] Sezer, S., The Hermite-Hadamard inequality for s-Convex functions in the third sense AIMS Mathematics, 6(7), (2021), 7719–7732.
21. [21] Tinaztepe, G. Yesilce, I., Adilov, G., Separation of $B^{-1}$-convex sets by $B^{-1}$-measurable Maps, Journal of Convex Analysis, 21(2)(2014), 571–580.
22. [22] Yeşilce, İ., Adilov, G., Hermite-Hadamard inequalities for $B$-convex and $B^{-1}$-convex functions. International Journal of Nonlinear Analysis and Applications, 8(1)(2017), 225–233.
23. [23] Zhang, K.S., Wan, J.P., p-Convex functions and their properties, Pure and Applied Mathematics, 1(23)2007), 130–133.