Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new inequalities help in finding the error bounds for different numerical integration formulas in multiplicative calculus. The use of s-convex function extends the results for convex functions and covers a large class of functions, which is the main motivation for using s-convexity. To prove the inequalities, we derive two different integral identities for multiplicative differentiable functions in the setting of multiplicative calculus. Then, with the help of these integral identities, we prove some integral inequalities of the Simpson and Ostrowski types for multiplicative generalized convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities, to show the validity of the results for multiplicative s-convex functions. We also give some applications to quadrature formula and special means of real numbers within the framework of multiplicative calculus.

Fractional calculus extends the differentiation and integration of functions to non-integer order. This work presents a new identity to represent specific differentiable mappings through generalized fractional integrals. On the basis of the newly established identity, several Boole?s type inequalities for dif-ferentiable generalized convex functions are obtained. Generalized fractional integrals are more versatile as compared to the traditional integral operators because they embody them. This method brings the con-nection between integer order calculus and fractional calculus, which gives more effective tools for solving non-singular problems where integer order calculus tools can be ineffective. The results provide further understanding of geometric properties of differentiable mappings and generalized convex functions which give rise to new identities and inequalities enriching the field of integral inequalities. Some numerical examples and applications are given in order to support these results as more feasible and relevant.

In this paper, quantum Weddle?s type inequalities for convex functions have been derived, involving the extension of the classical results in the framework of q-calculus. We present new conditions that describe the behaviour of convex functions using the quantum calculus, which undermines the systematical theory of the related phenomena in theoretical and applied mathematics. We also investigate some aspects of these inequalities, such as scaling and translation, and demonstrate how some of the inequalities are connected to other existing inequalities in the intersection of convex analysis and optimization. Furthermore, numerical and graphical solutions to the inequalities applied to real-life problems are given, along with an illustration of the computation and the connection to the relevant inequalities. Therefore, the findings of the present work are also useful in extending the theory of convex functions.