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We establish new quantum Hermite-Hadamard and midpoint types inequalities via a parameter μ ∈ [0, 1] for a function F whose | α D q F| u is η-quasiconvex on [α, β] with u ≥ 1. Results obtained in this paper generalize, sharpen, and extend some results in the literature. For example, see (

We develop the Benkhettou-Hassani-Torres fractional (noninteger order) calculus on timescales by proving two chain rules for the α-fractional derivative and five inequalities for the α-fractional integral. The results coincide with well-known classical results when the operators are of (integer) order α = 1 and the timescale coincides with the set of real numbers. Mathematics Subject Classification

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The purpose of this work is to present the quantum Hermite-Hadamard inequality through the Green function approach. While doing this, we deduce some novel quantum identities. Using these identities, we establish some new inequalities in this direction. We contemplate the possibility of expanding the method, outlined herein, to recast the proofs of some known inequalities in the literature.