A new q-integral identity and estimation of its bounds involving generalized exponentially μ-preinvex functions

In the article, we introduce the generalized exponentially μ-preinvex function, derive a new q-integral identity for second order q-differentiable function, and establish several new q-trapezoidal type integral inequalities for the function whose absolute value of second q-derivative is exponentially μ-preinvex.


Preliminaries
In this section, we introduce the new definition of exponentially μ-preinvex function, establish a new q-integral identity, and obtain new associated q-bounds.
First of all, let K ⊂ n be a nonempty set, : K → be a continuous function, and μ : K × K → \ {0} and θ : K × K → n be two continuous bifunctions. Definition 2.1 A set K ⊆ n is said to be μ-invex with respect to the bifunctions μ(·, ·) and θ (·, ·) if a + kμ(b, a)θ (b, a) ∈ K for all a, b ∈ K and k ∈ [0, 1].
Note that the convex set with μ(b, a) = 1 and θ (b, a) = ba is an invex set, but the converse is not true. For example, the set K = \ (-1/2, 1/2) is an invex set with respect to θ and μ(b, a) = 1, where It is clear that K is not a convex set. Note that, if α = 0 or χ = 1, then the class of generalized exponentially μ-preinvex functions reduces to the class of generalized μ-preinvex functions. The class of generalized exponentially μ-preinvex function includes the class of of preinvexity for α = 0 and μ(b, a) = 1. Also note that if we take χ = e, then we have the class of exponentially μpreinvex functions, which is defined as follows.
Next, we recall some previously known concepts and results, which will be helpful in obtaining the quantum analogues of the main results of the article. Definition 2.4 (see [49,50]) Let 0 < q < 1 and : J = [a, b] → be an arbitrary function. Then the q-derivative of on J at t is defined as follows: We note that lim q→1 a D q (t) = d (t)/dt is just the classical derivative if is differentiable.
Definition 2.5 (see [49,50]) Let : J = [a, b] → be an arbitrary function. Then the second-order q-derivative on the interval J is defined by Similarly, the higher order q-derivative on J can be defined by a D n q (t) = a D q a D n-1 q (t) .
Then the q-integral on J is defined by Note that if a = 0, then we have the classical q-integral, which is defined as follows: Lemma 2.2 (see [49,50] Definition 2.7 (see [51]) Let a ∈ and n ∈ N. Then the q-analogue of a is defined by Definition 2.8 (see [51]) Let k, p > 0. Then B q (k, p) is defined by For more details for q-calculus, we recommend the literature [52][53][54][55] to the readers.

Results and discussions
In this section, we present our main results of the article.
Lemma 3.1 Let 0 < q < 1 and : K → be an arbitrary function such that D 2 q is qintegrable on K. Then one has Proof We clearly see that Multiplying both sides of the above equality by q 2 μ 2 (b, a)θ 2 (b, a)/(1 + q), we get the required result.
Using the same idea as in the proof of Theorem 3.2, we can complete the proof.  + μ(b, a)θ (b, a)) q + 1 -1  μ(b, a)θ (b, a)   a+μ(b,a)θ(b,a) a (3.5) Proof Using Lemma 3.1, Hölder's inequality, and the given hypothesis of the theorem, we get This completes the proof.
Using the same idea as in the proof of Theorem 3.2, we complete the proof.

Conclusion
In this paper, we have defined the class of generalized exponentially μ-preinvex functions and derived a new generalized quantum integral identity. With the help of this auxiliary result, we have obtained some new estimates of the quantum bounds essentially using the class of generalized exponentially μ-preinvex functions. It is worth to mention here that if we take χ = e, then all of the main results reduce to the results for exponentially μ-preinvex functions. To the best of our knowledge, these results are new in the literature. Since the quantum calculus has wide applications in many mathematical areas, this new class of functions can be applied to obtain more results in convex analysis, special functions, quantum mechanics, optimization theory, mathematical inequalities and may stimulate further research in different areas of pure and applied sciences.