We establish some new extensions of Hermite–Hadamard inequality for fractional integrals and present several applications for the Beta-function.
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M. Alomari and M. Darus, “On the Hadamard’s inequality for log-convex functions on the coordinates,” J. Inequal. Appl., Article ID 283147 (2009), 13 p.
S. S. Dragomir, “Two map**s in connection to Hadamard’s inequalities,” J. Math. Anal. Appl., 167, 49–56 (1992).
S. S. Dragomir, “On the Hadamard’s inequality for convex on the coordinates in a rectangle from the plane,” Taiwan. J. Math., 5, No. 4, 775–788 (2001).
S. S. Dragomir and R. P. Agarwal, “Two inequalities for differentiable map**s and applications to special means of real numbers and to trapezoidal formula,” Appl. Math. Lett., 11, No. 5, 91–95 (1998).
S. S. Dragomir, Y.-J. Cho, and S.-S. Kim, “Inequalities of Hadamard’s type for Lipschitzian map**s and their applications,” J. Math. Anal. Appl., 245, 489–501 (2000).
L. Fejér, “Über die Fourierreihen, II,” in: Math. Naturwiss. Anz Ungar. Akad. Wiss. [in Hungarian], 24 (1906), pp. 369–390.
J. Hadamard, “ Étude sur les propriétés des fonctions entières en particulier d’une fonction considérée par Riemann,” J. Math. Pures Appl. (9), 58, 171–215 (1893).
S.-R. Hwang, K.-L. Tseng, and K.-C. Hsu, “New inequalities for fractional integrals and their applications,” Turk. J. Math., 40, 471–486 (2016).
S.-R. Hwang and K.-L. Tseng, “New Hermite–Hadamard-type inequalities for fractional integrals and their applications,” Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 112, 1211–1223 (2018).
S.-R. Hwang, K.-C. Hsu, and K.-L. Tseng, “Hadamard-type inequalities for Lipschitzian functions in one and two variables with their applications,” J. Math. Anal. Appl., 405, 546–554 (2013).
S.-R. Hwang, S.-Y. Yeh, and K.-L. Tseng, “Refinements and similar extensions of Hermite–Hadamard inequality for fractional integrals and their applications,” Appl. Math. Comput., 249, 103–113 (2014).
U. S. Kirmaci, “Inequalities for differentiable map**s and applications to special means of real numbers and to midpoint formula,” Appl. Math. Comput., 147, 137–146 (2004).
U. S. Kirmaci and M. E. Özdemir, “On some inequalities for differentiable map**s and applications to special means of real numbers and to midpoint formula,” Appl. Math. Comput., 153, 361–368 (2004).
M. Z. Sarikaya, E. Set, H. Yaldiz, and N. Başak, “Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities,” Math. Comput. Model., 57, 2403–2407 (2013).
K.-L. Tseng, S.-R. Hwang, and S. S. Dragomir, “Fejér-type inequalities (I),” J. Inequal. Appl., Article ID 531976 (2010), 7 p.
G.-S. Yang and K.-L. Tseng, “On certain integral inequalities related to Hermite–Hadamard inequalities,” J. Math. Anal. Appl., 239, 180–187 (1999).
G.-S. Yang and K.-L. Tseng, “Inequalities of Hadamard’s type for Lipschitzian map**s,” J. Math. Anal. Appl., 260, 230–238 (2001).
C. Zhu, M. Fečkan, and J.-R. Wang, “Fractional integral inequalities for differentiable convex map**s and applications to special means and a midpoint formula,” J. Amer. Math. Soc., 8, No. 2, 21–28 (2012).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 3, pp. 407–424, March, 2020.
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Hwang, SR., Yeh, SY. & Tseng, KL. On Some Hermite–Hadamard Inequalities for Fractional Integrals and their Applications. Ukr Math J 72, 464–484 (2020). https://doi.org/10.1007/s11253-020-01793-y
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DOI: https://doi.org/10.1007/s11253-020-01793-y