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Fractional Hermite-Hadamard-type inequalities for interval-valued functions. (English) Zbl 1437.26025

The Hermite-Hadamard inequality discovered by Hermite and Hadamard is one of the most well-established inequalities in the theory of convex functions (see [J. E. Pečarić et al., Convex functions, partial orderings, and statistical applications. Boston, MA etc.: Academic Press (1992; Zbl 0749.26004)]). The Hermite-Hadamard inequality for the Riemann-Liouville fractional integral, defined in [A. A. Kilbas et al., Theory and applications of fractional differential equations. Amsterdam: Elsevier (2006; Zbl 1092.45003)], was given by M. Z. Sarikaya et al. [Math. Comput. Modelling 57, No. 9–10, 2403–2407 (2013; Zbl 1286.26018)].
The authors define an interval-valued right-sided Riemann-Liouville fractional integral, based on the definition of interval-valued left-sided Riemann-Liouville fractional integral, given by V. Lupulescu [Fuzzy Sets Syst. 265, 63–85 (2015; Zbl 1361.26001)], and establish the Hermite-Hadamard inequality and Hermite-Hadamard-type inequalities for the interval-valued right-sided Riemann-Liouville fractional integral of a convex interval-valued function (see [K. Nikodem, Aequationes Math. 33, 46–56 (1987; Zbl 0639.39012)]).
The results are useful for researchers on inequalities for convex functions.

MSC:

26D15 Inequalities for sums, series and integrals
26A33 Fractional derivatives and integrals
26E25 Set-valued functions
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
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