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    Hermite-Hadamard-Mercer-Type Inequalities for Harmonically Convex Mappings
    (Mdpi, 2021) You, Xuexiao; Ali, Muhammad Aamir; Budak, Hüseyin; Reunsumrit, Jiraporn; Sitthiwirattham, Thanin
    In this paper, we prove Hermite-Hadamard-Mercer inequalities, which is a new version of the Hermite-Hadamard inequalities for harmonically convex functions. We also prove Hermite-Hadamard-Mercer-type inequalities for functions whose first derivatives in absolute value are harmonically convex. Finally, we discuss how special means can be used to address newly discovered inequalities.
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    On some new midpoint inequalities for the functions of two variables via quantum calculus
    (Springer, 2021) You, Xuexiao; Ali, Muhammad Aamir; Erden, Samet; Budak, Hüseyin; Chu, Yu-Ming
    In this paper, first we obtain a new identity for quantum integrals, the result is then used to prove midpoint type inequalities for differentiable coordinated convex mappings. The outcomes provided in this article are an extension of the comparable consequences in the literature on the midpoint inequalities for differentiable coordinated convex mappings.
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    Quantum Inequalities of Hermite-Hadamard Type for r-Convex Functions
    (Hindawi Ltd, 2021) You, Xuexiao; Kara, Hasan; Budak, Hüseyin; Kalsoom, Humaira
    In this present study, we first establish Hermite-Hadamard type inequalities for r-convex functions via (q)(k2)-definite integrals. Then, we prove some quantum inequalities of Hermite-Hadamard type for product of two r-convex functions. Finally, by using these established inequalities and the results given by (Brahim et al. 2015), we prove several quantum Hermite-Hadamard type inequalities for coordinated r-convex functions and for the product of two coordinated r-convex functions.