In this paper, we use $(p,q)$-integral to establish some Fejér type inequalities. In particular, we generalize and correct existing results of quantum Fejér type inequalities by using new techniques and showing some problematic parts of those results. Most of the inequalities presented in this paper are significant extensions of results which appear in existing literatures.

In this paper, we apply $(p,q)$-calculus to establish some new Chebyshev-type integral inequalities for synchronous functions. In particular, we generalize results of quantum Chebyshev-type integral inequalities by using $(p,q)$-integral. By taking $p=1$ and $q\to 1$, our results reduce to classical results on Chebyshev-type inequalities for synchronous functions. Furthermore, we consider their relevance with other related known results.

In this paper, we use $q_a$- and $q^b$-integrals to establish some quantum Hermite–Hadamard–Fejér-type inequalities for $\eta$-convex functions. By taking $q\to 1$, our results reduce to classical results on Hermite–Hadamard–Fejér-type inequalities for $\eta$-convex functions. Moreover, we give some examples for quantum Hermite–Hadamard–Fejér-type inequalities for $\eta$-convex functions. Some results presented here for $\eta$-convex functions provide extensions of others given in earlier works for convex and $\eta$-convex functions.