Different types of quantum integral inequalities via \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\alpha ,m)$\end{document}(α,m)-convexity

In this paper, based on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\alpha,m)$\end{document}(α,m)-convexity, we establish different type inequalities via quantum integrals. These inequalities generalize some results given in the literature.


Introduction and preliminaries
Throughout the paper, let I := [a, b] ⊆ R with 0 ≤ a < b be an interval, I • be the interior of I and let 0 < q < 1 be a constant.
In 2014, Tariboon and Ntouyas defined the q-derivative and q-integral as follows.  ∞ n=0 q n f q n x + 1q n a for x ∈ I. Moreover, if c ∈ (a, x), then the q-integral on I is defined as In the same paper, they also proved the following q-Hölder inequality. holds for all x ∈ I and r 1 , r 2 > 1 with r -1 1 + r -1 2 = 1.
In 2018, Alp et al. generalized the Hermite-Hadamard inequality to the form of qintegrals as follows.
In 1993, Miheşan gave the definition of (α, m)-convex functions as follows.
This paper aims to establish different types of quantum integral inequalities via (α, m)convexity. Some relevant connections of the results obtained in this paper with previous ones are also pointed out.

Auxiliary results
For proving main results, we need the following lemma. Lemma 2.1 Let f : I → R be a continuous and q-differentiable function on I • with 0 < q < 1. Then the identity Proof By an identical transformation, we get Utilizing the above calculation and Definition 1.2, we have (2.5) (ii) Putting μ = 1, we have (2.8) Specially, taking μ = 1 1+q , we obtain the midpoint-like integral identity which is presented by Alp et al. in [2,Lemma 11].
It is worth to mention here that to the best of our knowledge the obtained identities (2.5)-(2.13) are new in the literature.
Next we provide some calculations which will be used in this paper.
When (λ + q)μ > λ, making use of Lemma 2.2 again, we get Similarly, we also get This completes the proof.
The following results of Lemma 2.4, Lemma 2.5 and Lemma 2.6 are stated without proof.

Main results
In 2018, Alp et al. established the q-Hermite-Hadamard inequality in [2]. Here we give a new proof, which is more concise.
Proof It is obvious that ∞ n=0 (1q)q n = 1, 0 < q < 1. Since Jensen's inequality defined on convex sets for infinite sums still remains true, utilizing this fact and Definition 1.2, we have Using Definition 1.2 and the convexity of f , we get The proof is completed.

3)
and 9 (λ, μ, α) Proof From Lemma 2.1, utilizing the property of the modulus and the (α, m)-convexity of | a D q f |, we have Similarly, we get Remark 3.1 Consider Corollary 3.1.
(i) Putting λ = 0, we get the midpoint-like integral inequality where a D q f a m and Specially, taking α = 1 = m, we obtain which is established by Alp et al. in [2,Theorem 13].
If | a D q f | r for r > 1 is (α, m)-convex, then the following theorem can be obtained.

Conclusions
In the present research, based on a new quantum integral identity with multiple parameters, we have developed some quantum error estimations of different type inequalities through (α, m)-convexity, such as the midpoint-like inequalities, the Simpson-like inequalities, the averaged midpoint-trapezoid-like inequalities and the trapezoid-like inequalities. The inequalities derived in this work are very helpful in error estimations involved in various approximation processes. We expect that the ideas of this article will facilitate further study concerning quantum integral inequalities.