In this work, we prove a generalized version of Hermite-Hadamard-Mercer type inequalities using the Beta function. Moreover, we prove some new trapezoidal type inequalities involving Beta functions for differentiable convex functions. The main advantage of these inequalities is that these can be converted into similar classical integral inequalities, Riemann-Liouville fractional integral inequalities and $k$-Riemann-Liouville fractional integral inequalities. Finally, we give applications to special means of real numbers for newly established inequalities.

In this study, we use quantum calculus to prove Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in the second sense. The newly proven results are also shown to be an extension of comparable results in the literature, like the results of [1, 12, 16]. Furthermore, it is provided that how the newly discovered inequalities can be applied to special means of real numbers.

In this paper, we prove a new identity involving the second derivative of the function and Riemann-Liouville fractional integrals. The newly established identity is then used to establish some new Simpson's type inequalities for twice differentiable convex functions. Finally, we give applications of special functions using the newly proved inequalities.

In this research, for interval-valued functions, we give a new version of Jensen inclusion which is called Jensen-Mercer inclusion. Moreover, we establish some new inclusions of Hermite-Hadamard-Mercer type for interval-valued functions.

In this work, we prove Ostrowski-Mercer inequalities for differentiable harmonically convex functions. It is also shown that the newly proved inequalities can be converted into some existing inequalities. Furthermore, it is provided that how the newly discovered inequalities can be applied to special means of real numbers.

In this work, we introduce the concept of double quantum integrals for the interval-valued functions of two variables. We offer several new inclusions of the Hermite-Hadamard type for co-ordinated convex interval-valued functions using the newly defined integrals. Moreover, we prove trapezoidal type inequalities for interval-valued functions of two variables using the ideas of the Pompeiu--Hausdorff distance between the intervals. It is also revealed that the results offered in this work are the generalization of several existing results.

In this work, two generalized quantum integral identities are proved by using some parameters. By utilizing these equalities we present several parameterized quantum inequalities for convex mappings. These quantum inequalities generalize many of the important inequalities that exist in the literature, such as quantum trapezoid inequalities, quantum Simpson's inequalities and quantum Newton's inequalities. We also give some new midpoint type inequalities as special cases. The results in this work naturally generalize the results for the Riemann integral.

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 paper, we first obtain an identity for twice partially differentiable mappings involving some parameters. Moreover, by utilizing this identity and functions whose twice partially derivatives in absolute value are co-ordinated convex, we establish some inequalities which generalize several inequalities, such as trapezoid, midpoint and Simpson's inequalities.

In this paper, we prove two identities involving quantum derivatives, quantum integrals, and certain parameters. Using the newly proved identities, we prove new inequalities of Simpson's and Newton's type for quantum differentiable convex functions under certain assumptions. Moreover, we discuss the special cases of our main results and obtain some new and existing Simpson's type inequalities, Newton's type inequalities, midpoint type inequalities and trapezoidal type inequalities.

In this note, for differentiable convex functions, we prove some new Ostrowski-Mercer inequalities. These inequalities generalize an Ostrowski inequality and related inequalities proved in [3,5]. Some applications to special means are also given.

In this study, we give the notions about some new post-quantum partial derivatives and then use these derivatives to prove an integral equality via post-quantum double integrals. We establish some new post-quantum Ostrowski type inequalities for differentiable coordinated functions using the newly established equality. We also show that the results presented in this paper are the extensions of some existing results.