In this paper, quantum version of Montgomery identity by applying $q^{b}$-integral is obtained. By employing the established identity, some new quantum Ostrowski type inequalities are proved. Several special cases of our main results are presented. It is also shown that the results presented in this paper generalize several other well known inequalities given in the existing literature on this subject.

In this article, we derive Hermite-Hadamard inequalities for preinvex functions using the quantum integrals and show their validation with mathematical examples. We prove midpoint and trapezoidal inequalities for q^{ϰ₂}-differentiable preinvex functions via q^{ϰ₂}-quantum integrals. Moreover, we show with an example that the already proved inequality of Hermite-Hadamard type for preinvex functions via q_{ϰ₁}-quantum integrals is not valid for preinvex functions and we give its correct version. We prove the midpoint inequalities for q_{ϰ₁}-differentiable preinvex functions via q_{ϰ₁}-quantum integrals. It is also shown that the newly proved results transformed into some known results by considering the limit q→1⁻ and η(ϰ₂,ϰ₁)=-η(ϰ₁,ϰ₂)=ϰ₂-ϰ₁ in the newly derived results.

In this research, we offer two new quantum integral equalities for recently defined $q^{b}$-integral and derivative, the derived equalities then used to prove quantum integral inequalities of Simpson’s and Newton’s type for preinvex functions. We also considered the special cases of established results and offer several new and existing results inside the literature of Simpson’s and Newton’s type inequalities.

In this paper first we present some new identities by using the notions of quantum integrals and derivatives which allows us to obtain new quantum Simpson’s and quantum Newton’s type inequalities for differentiable convex functions by using the q_{x}-quantum integral and q^{y}-quantum integral. In particular, this paper generalises and extends previous results obtained by the various authors in the field of quantum and classical integral inequalities.

In this paper, we introduce the notion of generalized fractional integrals for the interval-valued functions of two variables. We establish Hermite-Hadamard type inequalities and some related inequalities for co-ordinated convex interval-valued functions by using the newly defined integrals. It is also proved that the results given in this paper are the strong generalization of already published ones.

In this investigation, we introduce new definitions of (p,q)-derivative and integral and discuss their fundamental properties. Some new Hermite-Hadamard inequalities employing newly defined (p,q)-integral are also obtained for convex functions. Furthermore, we are interested in finding (p,q)-estimates for midpoint and trapezoidal type inequalities for differentiable functions. It is revealed that the inequalities given in this article are extensions of offered inequalities in the literature of Hermite-Hadamard inequalities.

In this article, by using the notion of newly defined $q_{1}q_{2}$-derivatives and integrals some new Simpson’s type inequalities for co-ordinated convex functions are shown. The outcomes raised in this paper are extensions and generalizations of the comparable results in the literature on Simpson’s inequalities for co-ordinated convex functions.