Quantum analog of some Hermite-Hadamard-type inequalities for left-right 𝑞-differentiable convex functions

The main purpose of this study is to obtain the quantum analogue of some classical Hermite-Hadamard-type inequalities. Firstly, the concepts of right quantum derivative and definite right quantum integral, similar to the left quantum derivative and definite left quantum integral on the interval [𝑎, 𝑏] of real numbers are introduced and investigated. Then the quantum analogues of some classical integral identities are established through a combination of left and right quantum integrals. Using these identities, some trapezoid, midpoint, and generalized Hermite-Hadamard types inequalities are obtained for the functions whose first-order left and right quantum derivatives in absolute value at certain powers are convex functions. As specific cases of the generalized Hermite-Hadamard type inequalities, the 𝑞-versions of some Bullen, Simpson, and weighted Bullen type inequalities are derived. Finally, quantum analogues of the general versions of Hölder-İşcan and improved power-mean inequalities are found, and the generalized and improved forms of the trapezoid and midpoint type inequalities related to left and right quantum integrals are derived by applying these inequalities. Furthermore, the obtained results are also discussed, when 𝑞 → 1−.