Simpson-type inequalities for higher order differentiable functions and applications

Integral inequalities are one of the most important tools used in both theoretical and applied mathematics. In some problems, the exact value of the integral cannot be calculated. In such cases it is necessary to develop approximation methods. Therefore, some mathematicians have studied integral inequalities for various classes of functions. Hermite-Hadamard, Simpson, Ostrowski, Ostrowski, Chebyshev, Grüss and Ostrowski-Grüss inequalities are some of the important inequalities in the literature. For example, Simpson-type integral inequalities play a critical role in mathematical analysis and in solving precision and reliability problems in numerical integration methods. Simpson's Rules, introduced by Thomas Simpson, are important approximation methods used in numerical integration. The simplest of these is the two-core model known as the Simpson 1/3 formula. Simpson's 3/8 rule, also known as Simpson's second formula or Newton's formula, is another approximation method introduced by Thomas Simpson. These approximation methods give the approximate value of an integral. One of the most effective methods used to determine the boundaries of the difference between approximation methods and integrals is integral inequalities. Accordingly, integral inequalities based on Simpson's rules are called Simpson-type inequalities. Recently, many researchers have found many results related to Simpson's inequalities. In this context, more precise results, equivalents and generalised versions of the classical Simpson inequality as well as new Simpson-type inequalities under different assumptions of functions have been studied. In this thesis, in the light of the ongoing work on Simpson-type inequalities based on convex functions, Simpson-type inequalities for higher order differentiable functions will be studied. Firstly, an integral identity involving higher order differentiable functions will be established with the help of the dual kernel. Then, using this identity and convex function properties, Simpson-type integral inequalities will be found. In addition, new Simpson-type approximation methods that can be used in numerical integration will be developed with the help of the results obtained while searching for these inequalities. Furthermore, the relations between the approximations obtained from the Simpson-type results and the real values of the integrals will be analysed.