Pompeiu mean value theorem for M-fractional derivatives and applications to inequalities

The theory of integral inequalities is one of the most useful mathematical tools in both pure and applied mathematics. The exact value of an integral can not be calculated in some problems. In these cases, it is necessary to develop approximation methods. Therefore, some mathematicians worked on integral inequalities for various classes of functions. Hermite-Hadamard, Ostrowski, Chebyshev, Grüss and Ostrowski-Grüss are some of the important inequalities in literature. Recently, some researchers focused on Ostrowski type inequalities obtained by using classical mean value theorem and Pompeiu mean value theorem which is introduced by Pompeiu in 1946. After that, Ostrowski type inequalities based on Pompeiu mean value theorem for different classes of functions and their weighted versions are examined by many mathematicians. On the other hand, Fractional calculus has grown into a significant pillar for mathematical analysis in recent years. Fractional integrals and derivatives were not generally known for many researchers and up to recent years has been used only in a pure mathematical context. However, these integrals and derivatives have been used in many application areas of mathematics in the last few decades. This theory have especially become indispensable in some areas of sciences. For example, it is impossible to describe a viscoelastic process without using a fractional derivative. Because of the wide of the application areas, new fractional computation methods such as local, conformable, alternative and M-Fractional have been developed. In addition, a large number of results involving fractional derivatives and integrals has been obtained in inequality theory which is one of the most important application areas of mathematics. For instance, fundamental inequalities such as Hermite-Hadamard, Ostrowski, Grüss and Ostrowski-Grüss were established by using Conformable Fractional Calculus which is used to calculate the integral and derivative values in between 0 and 1. Furthermore, some researchers gave Pompeiu mean value theorem for conformable and local fractional theory and they also developed some inequalities by using fractional Pompeiu mean calue theorem. Inspired and motivated by ongoing research on Ostrowski type inequalities based on Pompeiu mean value theorem, Pompeiu mean value theorem for M-Fractional derivatives will be proved in this thesis. Aditionally, new Ostrowski type inequalities obtained by using M-Fractional Pompeiu mean value theorem wil be established. The weighted versions of these Ostrowski type inequalities will be obtained. Finally, some applications of the inequalities which will be developed in this thesis will be also given.