Abstract
In this paper, we first obtain a generalized integral identity for twice local fractional differentiable mappings on fractal sets \({\mathbb {R}}^{\alpha }\, (0<\alpha \le 1)\) of real line numbers. Then, using twice local fractional differentiable mappings that are in absolute value at certain powers generalized strongly m-convex, we obtain some new estimates on generalization of trapezium-like inequalities. We also discuss some new special cases which can be deduced from our main results.
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References
Agarwal, P.: Some inequalities involving Hadamard type \(k\)-fractional integral operators. Math. Methods Appl. Sci. 40, 3882–3891 (2017)
Agarwal, P.; Jleli, M.; Tomar, M.: Certain Hermite-Hadamard type inequalities via generalized \(k\)-fractional integrals. J. Inequal. Appl. 2017, 10 (2017)
Anastassiou, G.: Fractional Differentiation Inequalities. Springer, Heidelberg (2009)
Anastassiou, G.: Advances on Fractional Inequalities. Springer, New York (2011)
Anastassiou, G.: Intelligent Mathematics: Computational Analysis. Springer, Heidelberg (2011)
Budak, H.; Sarikaya, M.Z.; Yildirim, H.: New inequalities for local fractional integrals. RGMIA Res. Rep. Collect. 18, 13 (2015)
Choi, J.-S.; Set, E.; Tomar, M.: Certain generalized Ostrowski type inequalities for local fractional integrals. Commun. Korean Math. Soc. 32, 601–617 (2017)
Erden, S.; Sarikaya, M.Z.: Generalized Pompeiu type inequalities for local fractional integrals and its applications. Appl. Math. Comput. 274, 282–291 (2016)
Jumarie, G.: Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions. Appl. Math. Lett. 22, 378–385 (2009)
Lara, T.; Merentes, N.; Quintero, R.; Rosales, E.: On strongly \(m\)-convex functions. Math. Aeterna 5(3), 521–535 (2015)
Lara, T.; Merentes, N.; Quintero, R.: On inequalities of Fejér and Hermite-Hadamard types for strongly \(m\)-convex functions. Math. Aeterna 5(5), 777–793 (2015)
Mo, H.-X.: Generalized Hermite-Hadamard inequalities involving local fractional integral. Arxiv 2014, 8 (2014)
Mo, H.-X.; Sui, X.: Generalized \(s\)-convex functions on fractal sets. Abstr. Appl. Anal. 2014, 8 (2014)
Mo, H.-X.; Sui, X.: Hermite-Hadamard type inequalities for generalized \(s\)-convex functions on real linear fractal set \({\mathbb{R}}^{\alpha }\, (0<\alpha \le 1)\). Math. Sci. (Springer) 11, 241–246 (2017)
Mo, H.-X.; Sui, X.; Yu, D.-Y.: Generalized convex functions on fractal sets and two related inequalities. Abstr. Appl. Anal. 2014, 7 (2014)
Sarikaya, M.Z.; Budak, H.: Generalized Ostrowski type inequalities for local fractional integrals. Proc. Am. Math. Soc. 145, 1527–1538 (2017)
Sarikaya, M.Z.; Erden, S.; Budak, H.: Some generalized Ostrowski type inequalities involving local fractional integrals and applications. RGMIA Res. Rep. Collect. 18, 12 (2015)
Sarikaya, M.Z.; Tunç, M.; Budak, H.: On generalized some integral inequalities for local fractional integrals. Appl. Math. Comput. 276, 316–323 (2016)
Set, E.; Tomar, M.: New inequalities of Hermite-Hadamard type for generalized convex functions with applications. Facta Univ. Ser. Math. Inform. 31, 383–397 (2016)
Srivastava, H.M.; Choi, J.-S.: Zeta and \(q\)-Zeta functions and associated series and integrals. Elsevier Inc, Amsterdam (2012)
Tomar, M.; Agarwal, P.; Jleli, M.; Samet, B.: Certain Ostrowski type inequalities for generalized \(s\)-convex functions. J. Nonlinear Sci. Appl. 10, 5947–5957 (2017)
Yang, X.-J.: Generalized local fractional Taylor’s formula with local fractional derivative. Arxiv 2011, 5 (2011)
Yang, X.-J.: Local fractional functional analysis and its applications. Asian Academic publisher Limited, Hong Kong (2011)
Yang, X.-J.: Advanced Local Fractional Calculus and its Applications. World Science Publisher, New York (2012)
Yang, X.-J.: Local fractional Fourier analysis. Adv. Mech. Eng. Appl. 1, 12–16 (2012)
Yang, X.-J.: Local fractional integral equations and their applications. Adv. Comput. Sci. Appl. 1, 234–239 (2012)
Yang, Y.-J.; Baleanu, D.; Yang, X.-J.: Analysis of fractal wave equations by local fractional Fourier series method. Adv. Math. Phys. 2013, 6 (2013)
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Anastassiou, G., Kashuri, A. & Liko, R. Local fractional integrals involving generalized strongly m-convex mappings. Arab. J. Math. 8, 95–107 (2019). https://doi.org/10.1007/s40065-018-0214-8
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DOI: https://doi.org/10.1007/s40065-018-0214-8