New Hadamard-type inequalities via $(s,m_{1},m_{2})$-convex functions

B. Bayraktar   ; S. I. Butt   ; Sh. Shaokat   ; J. E. Nápoles Valdés

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 31 (2021) no. 4, pp. 597-612

Résumé

The article introduces a new concept of convexity of a function: $(s,m_{1},m_{2})$-convex functions. This class of functions combines a number of convexity types found in the literature. Some properties of $(s,m_{1},m_{2})$-convexities are established and simple examples of functions belonging to this class are given. On the basis of the proved identity, new integral inequalities of the Hadamard type are obtained in terms of the fractional integral operator. It is shown that these results give us, in particular, generalizations of a number of results available in the literature.

Détail

Keywords: convex function, Hadamard type inequality, Riemann-Liouville fractional integral, power mean inequality.
Mots-clés : Hölder inequality

Bibliographie

[1] Akdemir A. O., Butt S. I., Nadeem M., Ragusa M. A., “New general variants of Chebyshev type inequalities via generalized fractional integral operators”, Mathematics, 9:2 (2021), 122 | DOI

[2] Bayraktar B., “Some new inequalities of Hermite–Hadamard type for differentiable Godunova–Levin functions via fractional integrals”, Konuralp Journal of Mathematics, 8:1 (2020), 91–96 https://dergipark.org.tr/en/pub/konuralpjournalmath/issue/31494/585770 | MR

[3] Bayraktar B., “Some new generalizations of Hadamard-type midpoint inequalities involving fractional integrals”, Problemy Analiza — Issues of Analysis, 9(27):3 (2020), 66–82 | DOI | MR | Zbl

[4] Bayraktar B., “Some integral inequalities of Hermite–Hadamard type for differentiable $(s,m)$-convex functions via fractional integrals”, TWMS Journal of Applied and Engineering Mathematics, 10:3 (2020), 625–637

[5] Bayraktar B., Kudaev V. Ch., “Some new integral inequalities for $(s, m)$-convex and $(\alpha, m)$-convex functions”, Bulletin of the Karaganda University. Mathematics Series, 2019, no. 2(94), 15–25 | DOI | MR

[6] Breckner W. W., “Stetigkeitsaussagen f{ü}r eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen R{ä}umen”, Publications de l'Institut Mathématique. Nouvelle série, 23(37) (1978), 13–20 (in German) | Zbl

[7] Butt S. I., Nadeem M., Qaisar Sh., Akdemir A. O., Abdeljawad Th., “Hermite–Jensen–Mercer type inequalities for conformable integrals and related results”, Advances in Difference Equations, 2020:1 (2020), 501 | DOI

[8] Dragomir S. S., “On some new inequalities of Hermite–Hadamard type for $m$-convex functions”, Tamkang Journal of Mathematics, 33:1 (2002), 45–56 | DOI | MR

[9] Dragomir S. S., Fitzpatrick S., “The Hadamard inequality for $s$-convex functions in the second sense”, Demonstratio Mathematica, 32:4 (1999), 687–696 | DOI | MR | Zbl

[10] Gao Zh., Li M., Wang J., “On some fractional Hermite–Hadamard inequalities via $s$-convex and $s$-Godunova–Levin functions and their applications”, Boletín de la Sociedad Matemática Mexicana, 23:3 (2017), 691–711 | DOI | Zbl

[11] Guzmán P. M., Kórus P., Nápoles Valdés J. E., “Generalized integral inequalities of Chebyshev type”, Fractal and Fractional, 4:2 (2020), 10 | DOI | Zbl

[12] Kadakal H., “$(m_{1},m_{2})$-convexity and some new Hermite–Hadamard type inequalities”, International Journal of Mathematical Modelling and Computations, 9:4 (2019), 297–309

[13] Kadakal H., “$(\alpha,m_{1},m_{2})$-convexity and some inequalities of Hermite–Hadamard type”, Communications Faculty Of Science University of Ankara. Series A1. Mathematics and Statistics, 68:2 (2019), 2128–2142 | DOI | MR

[14] Khan Sh., Khan M. A., Butt S. I., Chu Y.-M., “A new bound for the Jensen gap pertaining twice differentiable functions with applications”, Advances in Difference Equations, 2020:1 (2020), 333 | DOI

[15] Mehmood N., Butt S. I., Pečarić Ð., Pečarić J., “Generalizations of cyclic refinements of Jensen's inequality by Lidstone's polynomial with applications in information theory”, Journal of Mathematical Inequalities, 14:1 (2020), 249–271 | DOI | MR | Zbl

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

[17] Nápoles Valdés J. E., Rodríguez J. M., Sigarreta J. M., “New Hermite–Hadamard type inequalities involving non-conformable integral operators”, Symmetry, 11:9 (2019), 1108 | DOI

[18] {Ö}zdemir 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

[19] Park J., “Generalization of Ostrowski-type inequalities for differentiable real $(s,m)$-convex mappings”, Far East Journal of Mathematical Sciences, 49:2 (2011), 157–171 | MR | Zbl

[20] Sarikaya M. Z., Aktan N., “On the generalization of some integral inequalities and their applications”, Mathematical and Computer Modelling, 54:9–10 (2011), 2175–2182 | DOI | MR | Zbl

[21] Set E., Butt S. I., Akdemir A., Karaoǧlan O., Abdeljawad Th., “New integral inequalities for differentiable convex functions via Atangana–Baleanu fractional integral operators”, Chaos, Solitons and Fractals, 143 (2021), 110554 | DOI | MR

[22] Toader G., “Some generalizations of the convexity”, Proceedings of the Colloquium on Approximation and Optimization, Cluj-Napoca University, Cluj-Napoca, 1985, 329–338 | Zbl

[23] Vivas-Cortez M. J., Hernàndez H. J. E., “Ostrowski and Jensen-type inequalities via $(s,m)$-convex functions in the second sense”, Boletín de la Sociedad Matemática Mexicana, 26:2 (2020), 287–302 | DOI | MR | Zbl