ArticleNo Access

AN IMPROVEMENT OF HÖLDER INTEGRAL INEQUALITY ON FRACTAL SETS AND SOME RELATED SIMPSON-LIKE INEQUALITIES

CHUNYAN LUO

Department of Mathematics, College of Science, China Three Gorges University, Yichang 443002, P. R. China

E-mail Address: luochunyanctgu@gmail.com

Search for more papers by this author

YUPING YU

Department of Mathematics, College of Science, China Three Gorges University, Yichang 443002, P. R. China

E-mail Address: yupingyuctgu@163.com

Search for more papers by this author

, and

TINGSONG DU

https://orcid.org/0000-0002-6072-1788

Department of Mathematics, College of Science, China Three Gorges University, Yichang 443002, P. R. China

E-mail Address: tingsongdu@ctgu.edu.cn

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0218348X21501267Cited by:18 (Source: Crossref)

Abstract

The purpose of this work is to investigate some inequalities for generalized $s$ $s$ -convexity on fractal sets $ℝ^{α}$ , which are associated with Simpson-like inequalities. To this end, an improved version of Hölder inequality and a Simpson-like identity on fractal sets are established, in view of which we give several estimation-type results involving Simpson-like inequalities for the first-order differentiable mappings. Moreover, we provide five examples to illustrate our results. As applications with respect to local fractional integrals, we derive two inequalities according to $α$ -type special means and generalized probability density functions.

Keywords:

References

1. D. S. Mitrinović, J. E. Pečarić and A. M. Fink , Classical and New Inequalities in Analysis (Kluwer Academic Publishers, Dordrecht, 1993). Crossref, Google Scholar
2. İ. İşcan , New refinements for integral and sum forms of Hölder inequality, J. Inequal. Appl. 2019 (2019) 304. Crossref, Web of Science, Google Scholar
3. W. W. Breckner , Stetigkeitsaussagen für eine klasse verallgemeinerter konvexer funktionen in topologischen linearen räumen, Publ. Inst. Math. 23(37) (1978) 13–20. Google Scholar
4. D. Karapınar, S. Turhan, M. Kunt and İ. İşcan , Some new fractional integral inequalities for $s$ -convex functions, J. Nonlinear Sci. Appl. 10 (2017) 4552–4563. Crossref, Google Scholar
5. M. A. Khan, Y. Chu, T. U. Khan and J. Khan , Some new inequalities of Hermite–Hadamard type for $s$ -convex functions with applications, Open Math. 15 (2017) 1414–1430. Crossref, Web of Science, Google Scholar
6. P. Kórus , An extension of the Hermite–Hadamard inequality for convex and $s$ -convex functions, Aequat. Math. 93 (2019) 527–534. Crossref, Web of Science, Google Scholar
7. S. Özcan and İ. İşcan , Some new Hermite–Hadamard-type inequalities for $s$ -convex functions and their applications, J. Inequal. Appl. 2019 (2019) 201. Crossref, Web of Science, Google Scholar
8. M. A. Latif , Inequalities of Hermite–Hadamard type for functions whose derivatives in absolute value are convex with applications, Arab. J. Math. Sci. 21(1) (2015) 84–97. Google Scholar
9. E. Set, A. O. Akdemir and M. E. Özdemir , Simpson-type integral inequalities for convex functions via Riemann–Liouville integrals, Filomat 31(14) (2017) 4415–4420. Crossref, Web of Science, Google Scholar
10. M. A. Latif , On some new inequalities of Hermite–Hadamard type for functions whose derivatives are $s$ -convex in the second sense in the absolute value, Ukrain. Math. J. 67(10) (2016) 1552–1571. Crossref, Web of Science, Google Scholar
11. Y. Shuang and F. Qi , Some integral inequalities for $s$ -convex functions, Gazi. Univ. J. Sci. 31(4) (2018) 1192–1200. Web of Science, Google Scholar
12. T. S. Du, Y. J. Li and Z. Q. Yang , A generalization of Simpson’s inequality via differentiable mapping using extended $(s, m)$ -convex functions, Appl. Math. Comput. 293 (2017) 358–369. Web of Science, Google Scholar
13. T. S. Du, M. U. Awan, A. Kashuri and S. S. Zhao , Some $k$ -fractional extensions of the trapezium inequalities through generalized relative semi- $(m, h)$ -preinvexity, Appl. Anal. 100(3) (2021) 642–662. Crossref, Web of Science, Google Scholar
14. K. C. Hsu, S. R. Hwang and K. L. Tseng , Some extended Simpson-type inequalities and applications, Bull. Iran. Math. Soc. 43(2) (2017) 409–425. Web of Science, Google Scholar
15. J. G. Liao, S. H. Wu and T. S. Du , The Sugeno integral with respect to $α$ -preinvex functions, Fuzzy Sets Syst. 379 (2020) 102–114. Crossref, Web of Science, Google Scholar
16. M. A. Noor, K. I. Noor and M. U. Awan , Simpson-type inequalities for geometrically relative convex functions, Ukrain. Math. J. 70(7) (2018) 1145–1154. Crossref, Web of Science, Google Scholar
17. X. J. Yang , Advanced Local Fractional Calculus and its Applications (World Science Publisher, New York, 2012). Google Scholar
18. J. Choi, E. Set and M. Tomar , Certain generalized Ostrowski-type inequalities for local fractional integrals, Commun. Korean Math. Soc. 32(3) (2017) 601–617. Web of Science, Google Scholar
19. H. X. Mo and X. Sui , Generalized $s$ -convex functions on fractal sets, Abstr. Appl. Anal. 2014 (2014) 254737. Google Scholar
20. H. Budak, M. Z. Sarikaya and E. Set , Generalized Ostrowski-type inequalities for functions whose local fractional derivatives are generalized $s$ -convex in the second sense, J. Appl. Math. Comput. Mec. 15(4) (2016) 11–21. Crossref, Web of Science, Google Scholar
21. A. Kılıçman and W. Saleh , On some inequalities for generalized $s$ -convex functions and applications on fractal sets, J. Nonlinear Sci. Appl. 10 (2017) 583–594. Crossref, Google Scholar
22. A. Kılıçman and W. Saleh , Some generalized Hermite–Hadamard-type integral inequalities for generalized $s$ -convex functions on fractal sets, Adv. Difference Equ. 2015 (2015) 301. Crossref, Web of Science, Google Scholar
23. H. X. Mo and X. Sui , Hermite–Hadamard-type inequalities for generalized $s$ -convex functions on real linear fractal set $ℝ^{α} (0 < α < 1)$ , Math. Sci. 11 (2017) 241–246. Crossref, Google Scholar
24. A. Akkurt, M. Z. Sarikaya, H. Budak and H. Yildirim , Generalized Ostrowski-type integral inequalities involving generalized moments via local fractional integrals, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 111(3) (2017) 797–807. Crossref, Web of Science, Google Scholar
25. M. Z. Sarikaya and H. Budak , Generalized Ostrowski-type inequalities for local fractional integrals, Proc. Amer. Math. Soc. 145(4) (2017) 1527–1538. Crossref, Web of Science, Google Scholar
26. M. Tomar, P. Agarwal, M. Jleli and B. Samet , Certain Ostrowski-type inequalities for generalized $s$ -convex functions, J. Nonlinear Sci. Appl. 10 (2017) 5947–5957. Crossref, Google Scholar
27. S. Erden and M. Z. Sarikaya , Generalized Pompeiu-type inequalities for local fractional integrals and its applications, Appl. Math. Comput. 274 (2016) 282–291. Web of Science, Google Scholar
28. S. Erden, M. Z. Sarikaya and N. Çelik , Some generalized inequalities involving local fractional integrals and their applications for random variables and numerical integration, J. Appl. Math. Stat. Inform. 12(2) (2016) 49–65. Crossref, Web of Science, Google Scholar
29. M. Z. Sarikaya, T. Tunc and H. Budak , On generalized some integral inequalities for local fractional integrals, Appl. Math. Comput. 276 (2016) 316–323. Crossref, Web of Science, Google Scholar
30. S. Iftikhar, P. Kumam and S. Erden , Newton’s-type integral inequalities via local fractional integrals, Fractals 28(3) (2020) 2050037. Link, Web of Science, Google Scholar
31. S. Iftikhar, S. Erden, P. Kumam and M. U. Awan , Local fractional Newton’s inequalities involving generalized harmonic convex functions, Adv. Difference Equ. 2020 (2020) 185. Crossref, Web of Science, Google Scholar
32. M. Krnić and P. Vuković , Multidimensional Hilbert-type inequalities obtained via local fractional calculus, Acta Appl. Math. 169(1) (2020) 667–680. Crossref, Web of Science, Google Scholar
33. O. Almutairi and A. Kılıçman , Integral inequalities for $s$ -convexity via generalized fractional integrals on fractal sets, Mathematics 8 (2020) 53. Crossref, Web of Science, Google Scholar
34. H. Budak, M. Z. Sarikaya and H. Yildirim , New inequalities for local fractional integrals, Iran. J. Sci. Technol. Trans. Sci. 41(4) (2017) 1039–1046. Crossref, Google Scholar
35. M. Vivas, J. Hernández and N. Merentes , New Hermite–Hadamard and Jensen-type inequalities for $h$ -convex functions on fractal sets, Rev. Colomb. Mat. 50(2) (2016) 145–164. Crossref, Google Scholar
36. C. Y. Luo, H. Wang and T. S. Du , Fejér–Hermite–Hadamard-type inequalities involving generalized $h$ -convexity on fractal sets and their applications, Chaos Solitons Fractals 131 (2020) 109547. Crossref, Web of Science, Google Scholar
37. G. Anastassiou, A. Kashuri and R. Liko , Local fractional integrals involving generalized strongly $m$ -convex mappings, Arab. J. Math. 8 (2019) 95–107. Crossref, Web of Science, Google Scholar
38. A. Kılıçman and W. Saleh , On product of generalized $s$ -convex functions and new inequalities on fractal sets, Malays. J. Math. Sci. 11 (2017) 87–105. Web of Science, Google Scholar
39. Z. Robles, J. E. Sanabria and R. V. Sánchez, On a generalization of strongly $η$ -convex functions via fractal sets, preprint (2020), arXiv:2004.06852. Google Scholar
40. K. Sayevand , Mittag–Leffler string stability of singularly perturbed stochastic systems within local fractal space, Math. Model. Anal. 24(3) (2019) 311–334. Crossref, Web of Science, Google Scholar
41. R. V. Sánchez C. and J. E. Sanabria , Strongly convexity on fractal sets and some inequalities, Proyeccion.-J. Math. 39(1) (2020) 1–13. Crossref, Google Scholar
42. W. B. Sun , On generalization of some inequalities for generalized harmonically convex functions via local fractional integrals, Quaestiones Math. 42(9) (2019) 1159–1183. Crossref, Web of Science, Google Scholar
43. T. S. Du, H. Wang, M. A. Khan and Y. Zhang , Certain integral inequalities considering generalized $m$ -convexity on fractal sets and their applications, Fractals 27(7) (2019) 1950117. Link, Web of Science, Google Scholar
44. H. K. Jassim , Analytical approximate solutions for local fractional wave equations, Math. Methods Appl. Sci. 43(2) (2020) 939–947. Crossref, Web of Science, Google Scholar
45. S. Maitama and W. D. Zhao , Local fractional homotopy analysis method for solving non-differentiable problems on Cantor sets, Adv. Difference Equ. 2019 (2019) 127. Crossref, Web of Science, Google Scholar
46. J. Singh, H. K. Jassim and D. Kumar , An efficient computational technique for local fractional Fokker Planck equation, Physica A 555 (2020) 124525. Crossref, Web of Science, Google Scholar
47. X. J. Yang, F. Gao and H. M. Srivastava , Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations, Comput. Math. Appl. 73(2) (2016) 203–210. Crossref, Web of Science, Google Scholar
48. X. J. Yang, F. Gao and H. M. Srivastava , A new computational approach for solving nonlinear local fractional PDEs, J. Comput. Appl. Math. 339 (2018) 285–296. Crossref, Web of Science, Google Scholar
49. K. J. Wang , On a high-pass filter described by local fractional derivative, Fractals 28(3) (2020) 2050031. Link, Web of Science, Google Scholar
50. J. Zhang, Z. L. Pei, G. C. Xu, X. H. Zou and F. Qi , Integral inequalities of extended Simpson type for $(α, m)$ - $𝜀$ -convex functions, J. Nonlinear Sci. Appl. 10 (2017) 122–129. Crossref, Google Scholar