Hermite-Hadamard Inequalities for a New Class of Generalized-(s,m) via Fractional Integral

Abstract

This paper defines a new generalized (s,m)-σ convex function using the σ convex functions and provides some applications and exact results for this kind of functions. The new definition of the (s,m)-σ convex function class is used to obtain the Hermite Hadamard type integral inequalities existing in the literature, and new integral inequalities are obtained with the help of the σ-Riemann-Liouville fractional integral. Additionally, a new Hermite-Hadamard type fractional integral inequality is constructed using the σ-Riemann-Liouville fractional integral.

Keywords

• Hermite–Hadamard inequality
• fractional İntegral operator
• Convex function
• σ-convex function
• (s-m)-σ convex function.

Kaynakça

1. [Anderson G.D, Vamanamurthy M.K, Vuorinen M. Generalized convexity and inequalities. Journal of Mathematical Analysis and Applications. 2007; 335(2), 1294–1308.
2. Youness EA. E-convex sets, E-convex functions and E-convex programming. Journal of Optimization Theory and Applications. 1999; 102(2), 439–450.
3. Du TS, Li YJ, Yang ZQ. A generalization of Simpson’s inequality via differentiable mapping using extended (s, m)-convex functions. Applied Mathematics and Computation. 2017; 293, 358–369.
4. Wu S, Awan MU, Noor MA, Iftikhar K. On a new class of convex functions and integral inequalities. Journal of Inequalities and Applications. 2019; 131.
5. Mohammed PO, Abdeljawad T, Zeng S, Kashuri A. Fractional Hermite-Hadamard integral inequalities for a new class of convex functions. Symmetry. 2020; 12, 1485.
6. Park J. Generalization of Ostrowski–type inequalities for differentiable real (s,m)-convex mappings. Far East Journal of Mathematical Sciences. 2011; 49(2), 157-171.
7. Kilbas AA, Srivastava HM, Trujillo, JJ. Theory and Applications of Fractional Differential Equations; North-Holland Mathematics Studies, Volume 204. Elsevier Sci. B.V, Amsterdam, The Netherlands; 2006.
8. Osler TJ. The Fractional Derivative of a Composite Function. SIAM Journal on Mathematical Analysis 1970; 1, 288–293.

9. Sarikaya MZ, Set E, Yaldiz H, Başak N. Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities. Mathematical and Computer Modelling. 2013; 57, 2403–2407.
10. Akdemir AO, Özdemir ME, Ardıç MA, Yalçın A. Some new generalizations for GA -convex functions. Filomat, 2017; 31(4), 1009–1016.
11. Xu JZ, Raza U. Hermite–Hadamard Inequalities for Harmonic (s, m)-Convex Functions. Mathematical Problems in Engineering, Article ID1470837, 7 pages. 2020.
12. Sahoo SK, Tariq M, Ahmad, H, Kodamasingh B, Shaikh, AA, Botmart T, El-Shorbagy MA. Some Novel Fractional Integral Inequalities over a New Class of Generalized Convex Function. Fractal and Fractional, 2022; 6, 42.
13. Dragomir SS, Pecaric J, Persson LE. Some inequalities of Hadamard type. Soochow Journal of Mathematics. 1995; 21(3), 335–341.
14. Beckenbach EF. Convex functions. Bulletin of the American Mathematical Society. 1948; 54(5), 439–461.
15. Mitrinović DS, Lacković IB. Hermite and convexity. Aequationes Mathematicae. 1985; 28(1), 229–232.
16. Dragomir SS, Fitzpatrick S. The Hadamard inequalities for s-convex functions in the second sense. Demonstratio Mathematica. 1999; 32, 687–696.
17. Akdemir, AO, Dutta, H, Yüksel, E, Deniz, E. Inequalities for m-Convex Functions via ψ -Caputo Fractional Derivatives. Matematical Methods and Modelling in Applied Sciences, 2020; Vol. 123, 215-224, Springer Nature Switzerland.
18. Deniz, E, Akdemir, AO, Yüksel, E. New Extensions of Chebyshev-Pòlya-Szegö Type Inequalities via Conformable Integrals. AIMS Mathematics, 2019; 4(6), 1684-1697.