Document Type : Research Paper

Author

Bayburt University, Faculty of Applied Sciences, Department of Customs Management, Baberti Campus, 69000, Bayburt-Turkey.

Abstract

In this paper, we study the concept of exponential convex functions with respect to $s$ and prove Hermite-Hadamard type inequalities for the newly introduced this class of functions. In addition, we get some refinements of    the Hermite-Hadamard (H-H) inequality for functions whose first derivative in absolute value, raised to a certain power which is greater than one, respectively at least one, is exponential convex with respect to $s$. Our results coincide with the results obtained previously in special cases.

Keywords

Main Subjects

1. M. Bombardelli and S. Varošanec, Properties of h-convex functions related to the Hermite-Hadamard-Fejér inequalities, Comput. Math. Appl., 58 (9) (2009), pp. 1869-1877.
2. S.I. Butt, A. Kashuri, M. Tariq, J. Nasir, A. Aslam, and W. Gao, n-polynomial exponential type p-convex function with some related inequalities and their applications, Heliyon., 6 (11) (2020).
3. S.S. Dragomir and R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11 (5) (1998), pp. 91-95.
4. S.S. Dragomir and CEM Pearce, Selected Topics on Hermite- Hadamard Inequalities and Its Applications, Science direct working paper, S1574-0358, (2003), 04.
5. S.S. Dragomir, J. Pečarić and l.e. Persson, Some inequalities of Hadamard Type, Soochow J. Math., 21 (3) (2021), pp. 335-341.
6. J. Hadamard, Étude sur les propriétés des fonctions entières en particulier d’une fonction considérée par Riemann, J. Math. Pures Appl., 58 (1893), pp. 171-215.
7. İ. İşcan, New refinements for integral and sum forms of Hölder inequality, J. Inequal. Appl., 2019 (1) (2019), pp. 1-11.
8. İ. İşcan and M. Kunt, Hermite-Hadamard-Fejer type inequalities for quasi-geometrically convex functions via fractional integrals. J. Math., 2016 (2016).
9. İ. İşcan, H. Kadakal and M. Kadakal, Some new integral inequalities for n-times differentiable quasi-convex functions. Sigma., 35 (3) (2017), pp. 363-368.
10. H. Kadakal, Multiplicatively P-functions and some new inequalities, Ntmsci., 6 (4) (2018), pp. 111-118.
11. H. Kadakal, Hermite-Hadamard type inequalities for trigonometrically convex functions, Sci. Stud. Res. Ser. Math. Inform., 28 (2) (2019), pp. 19-28.
12. H. Kadakal, New Inequalities for Strongly r-Convex Functions, J. of Funt. Spaces., 2019 (2019).
13. M. Kadakal and İ. İşcan, Exponential type convexity and some related inequalities, J. Inequal. Appl., 2020 (2020), pp. 1-9.
14. M. Kadakal, İ. İşcan, P. Agarwal and M. Jleli, Exponential trigonometric convex functions and Hermite-Hadamard type inequalities, Math. Slovaca., 71 (1) (2021), pp. 43-56.
15. M. Kadakal, İ. İşcan, H. Kadakal and K. Bekar, On improvements of some integral inequalities, Honam Math. J., 43 (3) (2021), pp. 441-452.
16. M. Kadakal, H. Kadakal and İ. İşcan, Some new integral inequalities for n-times differentiable s-convex functions in the first sense, TJANT., 5 (2) (2017), pp. 63-68.
17. H. Kadakal, M. Kadakal and İ. İşcan, Some new integral inequalities for n-times differentiable s-convex and s-concave functions in the second sense, Math. Stat., 5 (2) (2017), pp. 94-98.
18. H. Kadaka, M. Kadakal and İ. İşcan, New type integral inequalities for three times differentiable preinvex and prequasiinvex functions, Open J. Math. Anal., 2 (1) (2018), pp. 33-46.
19. S. Maden, H. Kadakal, M. Kadakal and İ. İşcan, Some new integral inequalities for n-times differentiable convex and concave functions, J. Nonlinear Sci. Appl., 10 (12) (2017), pp. 6141-6148.
20. S. Özcan, Some Integral Inequalities for Harmonically (α, s)-Convex Functions, J. of Funt. Spaces., 2019 (2019).
21. S. Özcan and İ. İşcan, Some new Hermite-Hadamard type inequalities for s-convex functions and their applications, J. Inequal. Appl., 2009 (2019), pp. 1-11.
22. S. Varošanec, On h-convexity, J. Math. Anal. Appl., 326 (1) (2007), pp. 303-311.