On generalizations of integral inequalities

B. Bayraktar ; J. E. Nápoles ; F. Rabossi

Problemy analiza, Tome 11 (2022) no. 2, pp. 3-23.

Résumé

In the present study, several new generalized integral inequalities of the Hadamard and Simpson-type are obtained. The results were obtained for functions whose first and third derivatives are either convex or satisfy the Lipschitz condition or the conditions of the Lagrange theorem. In a particular case, these results not only confirm but also improve some upper bounds, well known in the literature for the Simpson and Hermite-Hadamard-type inequalities.

Détail

Keywords: convex function, Hermite–Hadamard inequality, Simpson-type inequality, Lipschitz conditions, Lagrange theorem, Riemann–Liouville fractional integral.

Bibliographie

[1] Alomari M., Darus M., Kirmaci, “U. S. Refinements of Hadamard-type inequalities for quasi-convex functions with applications to trapezoidal formula and to special means”, Comput. Math. with Appl., 59 (2010), 225–232 | DOI | MR | Zbl

[2] Alomari M., Hussain S., “Two inequalities for quasi-convex functions and applications”, Appl. Math. E-Notes, 11 (2011), 110–117 | MR | Zbl

[3] Awan M. U., Noor M. A., Mihai M. V., Noor K. I., “Fractional Hermite-Hadamard inequalities for differentiable s-Godunova-Levin functions”, Filomat, 30:12 (2016), 3235–3241 https://www.jstor.org/stable/10.2307/2489933 | DOI | MR | Zbl

[4] Bayraktar B., “Some integral inequalities for functions whose absolute values of the third derıvative is concave and $r$-convex”, Turkish J. Ineq., 4:2 (2020), 59–78

[5] Budak H., Tunç T., Sarıkaya M. Z., “On Hermite-Hadamard type inequalities for $\varphi$-convex function”, Miskolc Mathematical Notes, 20:1 (2019), 169–191 | DOI | MR | Zbl

[6] Butt S. I., Bayraktar B., Umar M., “Several new integral inequalities via $k$-Riemann–Liouville fractional integrals operators”, Probl. Anal. Issues Anal., 10:28-1 (2021), 3–22 | DOI | MR | Zbl

[7] Chun L., Qi F., “Inequalities of Simpson type for functions whose third derivatives are extended $s$-convex functions and applications to means”, J. Comp. Anal. Appl., 19:3 (2015), 555–569 | MR | Zbl

[8] Delavar M. R., Dragomir S. S., “Hermite–Hadamard's midpoint type inequalities for generalized fractional integrals”, RACSAM, 114:2 (2020), 73 | DOI | MR | Zbl

[9] 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 (1998), 91–95 | DOI | MR | Zbl

[10] Dragomir S. S., Agarwal R. P., Cerone P., “On Simpson's inequality and applications”, Journal of Inequalities and Applications, 5:6 (2000), 533–579 | MR | Zbl

[11] Hadamard J., “Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann”, J. Math. Pures Appl., 58 (1892), 171–215

[12] Hermite C. Sur deux limites d'une inté grale définie, Mathesis, 3:82 (1883)

[13] Hua J., Xi B.-Y., Qi F., “Some new inequalities of Simpson type for strongly $s$-convex functions”, Afrika Mat., 2014 | DOI | MR

[14] Hussain S., Qaisar S., “Generalizations of Simpson's type inequalities through preinvexity and prequasiinvexity”, Journal of Mathematics, 46:2 (2014), 1–9 | MR | Zbl

[15] Iqbal M., Bhatti M. I., Dragomir S. S., “Hadamard-Type Inequalities through $h$-convexity”, Research Report Collection, 14:63 (2011) | MR | Zbl

[16] Kashuri A., Meftah B., Mohammed P. O., “Some weighted Simpson type inequalities for differentiable $s$-convex functions and their applications”, Journal of Fractional Calculus and Nonlinear Systems, 1:1 (2021), 75–94 | DOI | MR

[17] Kirmaci U., “Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula”, Appl. Math. Comput., 147:5 (2004), 137–146 | DOI | MR | Zbl

[18] Latif M. A., “On some new inequalities of Hermite – Hadamard type for functions whose derivatives are $s$-convex in the Second Sense in the Absolute Value”, Ukrainian Mathematical Journal, 67:10 (2016), 1552–1571 | DOI | MR | Zbl

[19] Latif M. A., “Inequalities of Hermite – Hadamard type for functions whose derivatives in absolute value are convex with applications”, Arab Journal of Mathematical Sciences, 21:1 (2015), 84–97 | DOI | MR | Zbl

[20] Latif M. A., Kunt M. T., Dragomir S. S., Işcan I., “Post-quantum trapezoid type inequalities”, AIMS Mathematics, 5:4 (2020), 4011–4026 | DOI | MR | Zbl

[21] Matloka M., “Some inequalities of Simpson type for $h$-convex functions via fractional integrals”, Abstract and Applied Analysis, 2015 | DOI | MR

[22] Mihai M. V., Mitroi F-C., “Hermite – Hadamard type inequalities obtained via Riemann – Liouville fractional calculus”, Acta Math. Univ. Comenianae, 83:2 (2014), 209–215 | MR

[23] Nápoles J. E., Rabossi F., Samaniego A. D., Convex functions: Ariadne's thread or Charlotte's spiderweb?, Advanced Mathematical Models Applications, 5:2 (2020), 176–191

[24] Özdemir M. E., Butt S. I., Bayraktar B., Nasir J., “Several integral inequalities for $(\alpha, s, m)$-convex functions”, AIMS Mathematics, 5:4 (2020), 3906–3921 | DOI | MR | Zbl

[25] Delavar M. R., Kashuri A., De La Sen M., “On weighted Simpson's $\frac3 8$rule”, Symmetry, 13:1933 (2021)

[26] Sarıkaya M. Z., Set E., Özdemir M. E., “On new inequalities of Simpson's type for convex functions”, RGMIA Res. Rep. Coll., 13:2 (2010) | MR

[27] Sarıkaya M. Z., Yıldrım H., “On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals”, Miskolc Mathematical Notes, 17:2 (2017), 1049–1059 | DOI | MR

[28] Set E., Sami Karata S., Khan M. A., “Hermite-Hadamard type inequalities obtained via fractional integral for differentiable $m$-convex and $(\alpha, m)$-convex functions”, International Journal of Analysis, 2016 | MR