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Abstract Mathematical analysis has been considered the main part of mathematics for the last three centuries. The study of inequalities can be considered one of basics of mathematical analysis. In recent decades, many inequalities were developed and explored which were used together with their proofs in many applications in the Öelds of pure mathematics, applied mathematics and mathematical physics. An assembly of some of the proved and published inequalities are represented in Hardy, Littlewood and G. PÚlya [32] known as ìInequalitiesî which was published in 1934. We refer to ([13], and [54]), and the references cited there in [55] for more information. Fractional calculus which can be deÖned as the studying of di§erentiation and integra- tion of fractional order is one of resulting Öelds of the evolution of mathematical analysis, see ([35], [52], and [83]). There are many deÖnitions of fractional di§erentiation and integra- tion operators like RiemannñLiouville deÖnition, Caputo deÖnition, and Gr¸nwald-Letnikov deÖnition. The fractional calculus appeared in signal processing, temperature Öeld problem in oil strata, hydraulics of dams and di§usion problems, see ([16], [19], and [80]). In [38], the authors introduced a deÖnition of the fractional derivative called conformable fractional derivative. Later, in [1], the author proved some basic concepts of the conformable fractional calculus. Fractional inequalities generalized by many authors by using the conformable frac- tional calculus, like Ste§ensenís inequality [77], Copson and Converses Copsonís inequality [68], Hardyís inequality [71] and Hermite-Hadamardís inequalities ([4], [39], [40], [78], [81], and [82]). vi vii In this thesis, we will prove some fractional inequalities by utilizing conformable frac- tional calculus and consequently, we derive some classical integral inequalities as special cases of our main results. The thesis consists of four chapters arranged as follows: Chapter 1 contains an introduction to conformabe fractional calculus and some basic deÖnitions, lemmas and the necessary results of the conformable fractional calculus that are required for obtaining the main results in the next chapters. Chapter 2 consists of four sections. In the Örst section, we will introduce continuous Hardyís inequality and some of its generalizations. In the second section, we will present some generalizations of Yang and Hwangís and Pachpatteís inequalities. In the third section, we will present some new fractional Hardyís inequalities. In the fourth section, we present some applications of Hardyís inequality by using the conformable fractional calculus. Chapter 3 consists of three sections. In the Örst section, we will present an introduction to Opialís inequality and some of its extensions. In the second section, we will present some generalizations of the weighted fractional Opialís inequalities. In the third section, we will deduce some new fractional inequalities of Opialís inequalities. Chapter 4 consists of three sections. In the Örst section, we will provide an introduction to the Leindlerís inequality and some of its extensions. In the second section, we will present some new fractional Leindlerís inequalities by using conformable fractional calculus. In the third section, we will prove some new some new fractional reverse Leindlerís inequalities. |