Ifra Bashir Sial

Composite integral formulas offer greater accuracy by dividing the interval into smaller subintervals, which better capture the local behavior of function. In the finite volume method for solving differential equations, composite formulas are mostly used on control volumes to achieve high-accuracy solutions. In this work, error estimates of the composite Simpson’s formula for differentiable convex functions are established. These error estimates can be applied to general subdivisions of the integration interval, provided the integrand satisfies a first-order differentiability condition. To this end, a novel and general integral identity for differentiable functions is established by considering general subdivisions of the integration interval. The new integral identity is proved in a manner that allows it to be transformed into different identities for different subdivisions of the integration interval. Then, under the convexity assumption on the integrand, sharp error bounds for the composite Simpson’s formula are proved. Moreover, the well-known Hölder’s inequality is applied to obtain sharper error bounds for differentiable convex functions, which represents a significant finding of this study. Finally, to support the theoretical part of this work, some numerical examples are tested and demonstrate the efficiency of the new bounds for different partitions of the integration interval.

The main goal of the current study is to establish some new Milne’s rule type inequalities for single-time differentiable convex functions in the setting of quantum calculus. For this, we establish a quantum integral identity and then we prove some new inequalities of Milne’s rule type for quantum differentiable convex functions. These inequalities are very important in Open-Newton’s Cotes formulas because, with the help of these inequalities, we can find the bounds of Milne’s rule for differentiable convex functions in classical or quantum calculus. The method adopted in this work to prove these inequalities are very easy and less conditional compared to some existing results. Finally, we give some mathematical examples to show the validity of newly established inequalities.

In this research, we give a new version of Jensen inclusion for interval-valued functions, which is called Jensen-Mercer inclusion. Moreover, we establish some new inclusions of the Hermite-Hadamard-Mercer type for interval-valued functions. Finally, we give some applications of newly established inequalities to make them more interesting for the readers.

In this paper, we establish a new integral identity involving differentiable functions, and then we use the newly established identity to prove some Ostrowski–Mercer-type inequalities for differentiable convex functions. It is also demonstrated that the newly established inequalities are generalizations of some of the Ostrowski inequalities established inside the literature. There are also some applications to the special means of real numbers given.