Some new Hermite-Hadamard type inequalities via k-fractional integrals concerning differentiable generalized relative semi-(r; m, p, q, h₁ , h₂ ) -preinvex mappings

• Autorzy: Kashuri A. , Liko R.

Identyfikatory

DOI

10.1515/fascmath-2018-0004

• Języki publikacji: EN

Abstrakty

EN

In this article, we first presented a new identity concerning differentiable mappings defined on m-invex set via k-fractional integrals. By using the notion of generalized relative semi-(r; m,p, q, h₁, h₂ )-preinvexity and the obtained identity as an auxiliary result, some new estimates with respect to Hermite-Hadamard type inequalities via k-fractional integrals are established. It is pointed out that some new special cases can be deduced from main results of the article.

Słowa kluczowe

EN

Hermite-Hadamard type inequality; MT-convex function; Hölder's inequality; Minkowski inequality; power mean inequality; k-fractional integrals; m-invex

• Wydawca: Wydawnictwo Politechniki Poznańskiej
• Czasopismo: Fasciculi Mathematici
• Rocznik: 2018
• Tom: Nr 60
• Strony: 59--78
• Opis fizyczny: Bibliogr. 41 poz.

Twórcy

autor

Kashuri A.

• Department of Mathematics Faculty of Technical Science University Ismail Qemali Vlora,Albania

autor

Liko R.

• Department of Mathematics Faculty of Technical Science University Ismail Qemali Vlora, Albania

Bibliografia

• [1] Antczak T., Mean value in invexity analysis, Nonlinear Anal., 60(2005), 1473-1484.
• [2] Chen F., A note on Hermite-Hadamard inequalities for products of convex functions via Riemann-Liouville fractional integrals, Ital. J. Pure Appl. Math., 33(2014), 299-306.
• [3] Chen F.X., Wu S.H., Several complementary inequalities to inequalities of Hermite-Hadamard type for s-convex functions, J. Nonlinear Sci. Appl., 9(2)(2016), 705-716.
• [4] Chu Y.-M., Khan M.A., Ali T., Dragomir S.S., Inequalities for α-fractional differentiable functions, J. Ineąual. Appl., 2017(93), (2017), pp. 12.
• [5] Chu Y.M., Khan M.A., Khan T.U., Ali T., Generalizations of Hermite- Hadamard type inequalities for MT-convex functions, J. Nonlinear Sci. Appl., 9(5)(2016), 4305-4316.
• [6] Chu Y.-M., Wang M.-K., Optimal Lehmer mean bounds for the Toader mean, Results Math., 61(3-4)(2012), 223-229.
• [7] Chu Y.-M., Wang M.-K., Qiu S.-L., Optimal combinations bounds of root-square and arithmetic means for Toader mean, Proc. Indian Acad. Sci. Math. Sci., 122(1)(2012), 41-51.
• [8] Chu Y.M., Wang G.D., Zhang X.H., Schur convexity and Hadamard’s inequality, Math. Ineąual. Appl., 13(4)(2010), 725-731.
• [9] Dragomir S.S., Pečarić J., Persson L.E., Some inequalities of Hadamard type, Soochow J. Math., 21(1995), 335-341.
• [10] Du T.S., Liao J.G., Chen L.Z., Awan M.U., Properties and Riemann-Liouville fractional Hermite-Hadamard inequalities for the generalized (a, m)-preinvex functions, J. Ineąual. Appl., 2016(2016), Article ID 306, 24 pages.
• [11] Du T.S., Liao J.G., Li Y.J., Properties and integral inequalities of Hadamard-Simpson type for the generalized (s, m)-preinvex functions, J. Nonlinear Sci. Appl., 9(2016), 3112-3126.
• [12] Kashuri A,. Liko R., Generalizations of Hermite-Hadamard and Ostrowski type inequalities for MTm-preinvex functions, Proyecciones, 36(1)(2017), 45-80.
• [13] Kashuri A., Liko R., Hermite-Hadamard type inequalities for MTm-preinvex functions, Fasc. Math., 58(2017), 77-96.
• [14] Kashuri A., Liko R., Hermite-Hadamard type fractional integral inequalities for generalized (r; s, m, p)-preinvex functions, Eur. J. Pure Appl. Math., 10(3)(2017), 495-505.
• [15] Kavurmaci H., Avci M., Ozdemir M.E., New inequalities of Hermite-Hadamard type for convex functions with applications, arXiv:1006.1593v1 [math.CA], (2010), 1-10.
• [16] Khan M.A., Begum S., Khurshid Y., Chu Y.-M., Ostrowski type inequalities involving conformable fractional integrals, J. Ineąual. Appl., 2018(70), (2018), pp. 14.
• [17] Khan M.A., Chu Y.-M., KASHURI A., Liko R., Hermite-Hadamard type fractional integral inequalities for MTr;g m v)-preinvex functions, J. Comput. Anal. Appl., 26(8)(2019), 1487-1503.
• [18] Khan M.A., Chu Y.-M., Khan T.U., Khan J., Some new inequalities of Hermite-Hadamard type for s-convex functions with applications, Open Math., 15(2017), 1414-1430.
• [19] Khan M.A., Khurshid Y., Ali T., Hermite-Hadamard inequality for fractional integrals via nconvex functions, Acta Math. Univ. Comenianae, 79(1)(2017), 153-164.
• [20] Khan M.A., Khurshid Y., Ali T., Rehman N., Inequalities for three times differentiable functions, J. Math., Punjab Univ., 48(2)(2016), 35-48.
• [21] Liu W., Wen W., Park J., Hermite-Hadamard type inequalities for MT-convex functions via classical integrals and fractional integrals, J. Nonlinear Sci. Appl., 9(2016), 766-777.
• [22] Matłoka M., Inequalities for h-preinvex functions, Appl. Math. Comput., 234(2014), 52-57.
• [23] Mubeen S., Habibullah G.M., k-Fractional integrals and applications, Int. J. Contemp. Math. Sci., 7(2012), 89-94.
• [24] Omotoyinbo O., Mogbodemu A., Some new Hermite-Hadamard integral inequalities for convex functions, Int. J. Sci. Innovation Tech., 1(1)(2014), 1-12.
• [25] Pachpatte B.G., On some inequalities for convex functions, RGMIA Res. Rep. Coli., 6(2003).
• [26] Peng C., Zhou C., Du T.S., Riemann-Liouville fractional Simpson’s inequalities through generalized (m, h1, h2)-preinvexity, Ital. J. Pure Appl. Math., 38(2017), 345-367.
• [27] Pini R., Invexity and generalized convexity, Optimization, 22(1991), 513-525.
• [28] Qi F., Xi B.Y., Some integral inequalities of Simpson type for GA - e-convex functions, Georgian Math. J., 20(5)(2013), 775-788.
• [29] Qian W.-M., Chu Y.-M., Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters, J. Inequal. Appl., 2017(274), (2017), pp. 10.
• [30] Shi H.N., Two Schur-convex functions related to Hadamard-type integral inequalities, Publ. Math. Debrecen, 78(2)(2011), 393-403.
• [31] Tunç M., Göv E., Şanal Ü., On tgs-convex function and their inequalities, Facta Univ. Ser. Math. Inform., 30(5)(2015), 679-691.
• [32] Varošanec S., On h-convexity, J. Math. Anal. Appl., 326(1)(2007), 303-311.
• [33] Wang M.-K., Chu Y.-M., Landen inequalities for a class of hypergeometric functions with applications, Math. Inequal. Appl., 21(2)(2018), 521-537.
• [34] Wang M.-K., Qiu S.-L., Chu Y.-M., Infinite series formula for Hübner upper bound function with applications to Hersch-Pfluger distortion function, Math. Inequal. Appl., 21(3)(2018), 629-648.
• [35] Xi B.Y., Qi F., Inequalities of Hermite-Hadamard type for extended s-convex functions and applications to means, arXiv:1406.5409v1 [math.CA], (2014), 1-17.
• [36] Yang Z.-H., Qian W.-M., Chu Y.-M., Zhang W., On approximating the error function, Math. Inequal. Appl., 21(2)(2018), 469-479.
• [37] Yang Z.-H., Qian W.-M., Chu Y.-M., Zhang W., On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind, J. Math. Anal. Appl., 462(2)(2018), 1714-1726.
• [38] Yang Z.-H., Zhang W., Chu Y.-M., Sharp Gautschi inequality for parame- ter 0
• [39] Yang X.M., YangX.Q., Teo K.L., Generalized invexity and generalized invariant monotonicity, J. Optim. Theory Appl., 117(2003), 607-625.
• [40] Youness E.A., E-convex sets, E-convex functions, and E-convex programming, J. Optim. Theory Appl., 102(1999), 439-450.
• [41] Zhang X.M., Chu Y.M., Zhang X.H., The Hermite-Hadamard type inequality of GA-convex functions and its applications, J. Inequal. Appl., (2010), Article ID 507560, 11 pages.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).