Inequalities with fractional integrals

It is well known that in recent years, the theory of fractional analysis has evolved. Fractional integrals and derivatives, which are used in many application areas of mathematics, show that mathematical modeling of real objects and processes is adequate. Clear results cannot always be achieved in mathematical problems. In these cases, it is necessary to develop approach methods. At this point, inequality theory comes to the fore. For example, it is not easy to calculate fractional integrals and derivatives of many functions because the definitions of the Riemann-Liouville fractional integral and derivative depend on the integral. When the exact value of an integral cannot be calculated, the approximate value of the integral can be estimated with the help of inequalities. In such cases, inequality is needed. In this regard, integral inequalities have become among 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 approach methods. Therefore, some mathematicians have studied integral inequalities for various classes of functions. Hermite-Hadamard and Ostrowski are some of the important inequalities in literatüre. Within the scope of this thesis, in the light of the studies carried out on inequalities involving fractional integrals; Fractional integral inequalities for higher-order differentiable functions and their applications will be given. First of all, new identities involving Riemann-Liouville fractional integrals are presented by using higher-order differantiable functions. Afterwards, fractional integral inequalities for functions whose higher order derivatives are bounded are established. The error estimates of the quadratic formulas that arise when investigating these inequalities are also examined. Finally, applications for exponential functions, which are one of the most adaptive forms of the higher order functions considered, are presented