Quantum Hermite–Hadamard inequality by means of a Green function

It is known in the literature as the Hermite–Hadamard inequality. This inequality has instigated pletora of papers. Results concerning generalization, refinement, and extension of (1.1) are also found; see [1–9, 12, 14, 15, 17–20, 23, 27–30] and the references therein. In the early 16th century, the concept of q-calculus was introduced. Ever since, integral inequalities of the trapeziod, Ostrowski, Cauchy–Bunyakovsky–Schwarz, Grüss, Hölder, Grüss–C̆ebys̆ev, and other types have been established in the q-calculus sense. In 2014, Tariboon and Ntouyas [33] obtained the following q-calculus version of (1.1).

The aim of this work is to recast inequality (1.3) in Theorem 1.2 via another approach different from that presented in [11]. Specifically, we do this using a Green function. In the process, we establish some identities that are also used to obtain more results in this direction.
We organize this paper as follows. Section 2 contains a brief introduction of the quantum calculus. Our main results are then framed and proved in Sect. 3.

Preliminaries
Quantum calculus is known as the calculus without limits. In this section, we present a quick overview of the theory of q-calculus. The interested reader is invited to the book [16] for an in-depth study of this subject. We begin with these basic definitions. ( is called the q-derivative on [b 1 , b 2 ] of the function at w. In light of Definitions 2.1 and 2.2, we make the following remarks: 1. By taking b 1 = 0 expression (2.1) boils down to the well-known q-derivative D q ψ(w) of the function ψ(w) defined by Some known results in continuous calculus have been extended to the q-calculus framework as follows.
Then any ψ ∈ C 2 ([b 1 , b 2 ]) can be expressed as We now state and justify our main results.
Proof If we set x = qb 1 +b 2 q+1 in (3.1), then we get By computing we obtain that (3.4) Subtracting (3.4) from (3.3), we get: Next, we consider the function For this, the following cases are possible.
This implies that f is decreasing and f (b 1 ) = 0, which shows that f (u) ≤ 0. Thus f is also decreasing, and Hence f is increasing and f (b 2 ) = 0. So, f (u) ≤ 0 for all u ∈ [ qb 1 +b 2 q+1 , b 2 ]. Now, using (3.5) and the fact that ψ (u) ≥ 0 for all u ∈ [b 1 , b 2 ], since ψ is convex, we obtain the first inequality: For the right-hand side inequality, we recall that Subtracting (3.4) from (3.7), we get Then Here we also observe two cases.
. Combining these two cases, we conclude that F(u) ≥ 0 for all u ∈ [b 1 , b 2 ]. Applying (3.8) and the convexity of ψ, we establish the right-hand side of the desired inequality. The proof is complete.
Next, we prove new quantum Hermite-Hadamard inequalities for the class of monotone and convex functions.
) be such that |ψ | is a convex function. Then for any q ∈ (0, 1), the following inequality holds: Proof Using (3.5), we get: