The aim of this work is to develop quantum estimates for q-Simpson type integral inequalities for co-ordinated convex functions by using the notion of newly defined q₁q₂-derivatives and integrals. For this, we establish a new identity including quantum integrals and quantum numbers via q₁q₂- differentiable functions. After that, with the help of this equality, we achieved the results we want. 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.

In this article, we introduce a new concept of quantum integrals which is called $^{\kappa_{2}}T_{q}$-integral. Then we prove several properties of this concept of quantum integrals. Moreover, we present several Hermite-Hadamard type inequalities for $^{\kappa_{2}}T_{q}$-integral by utilizing differentiable convex functions. The results presented in this article are unification and generalization of the comparable results in the literature.

With respect to the non-integro-fractional derivative, in previous studies, the non-integro-fractional derivative of non-negative real numbers can be calculated. However, by previous denitions, the non-integro-fractional de- rivative of negative values can not be calculated due to t; 2 (0; 1). For example, (2)12 =2 R for t = 2 and = 1 2 : So what should we do for the non-integro-fractional derivative of “negative” real numbers? The pur- pose of this paper is to introduce more general derivative denition and we claim that we will obtain non-integro-fractional derivative of “all” real num- bers. Classic derivative, q-derivative, (p; q)-derivative, comformable fractional derivative, Katugampola fractional derivative and backward-forward dierence operator in Time Scale are the special cases of these general derivative deni- tions. These new denitions of ours must give us derivatives on both discrete and continuous calculus.

In this paper, we have proved the indispensable inequalities of classical analysis for interval value functions: Hölder, Cauchy-Schwarz and Carlson inequalities. Additionally, we achieved generalization for Carlson’s inequality for interval-valued functions.