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Abstract Historically, mathematical analysis has been the major and signi cant branch of math- ematics for the last three centuries. Indeed, inequalities became the heart of mathematical analysis. Many great mathematicians have made signi cant contributions to many new de- velopments of the subject, which led to the discovery of many new inequalities with proofs and useful applications in many elds of mathematical physics, pure and applied mathemat- ics. Indeed, mathematical inequalities became an important branch of modern mathematics in twentieth century through the pioneering work entitled Inequalities by G. H. Hardy, J. E. Littlewood and G. Pòlya [39], which was rst published treatise in 1934. This unique publication represents a paradigm of precise logic, full of elegant inequalities with rigorous proofs and useful applications in mathematics. In recent years the study of dynamic in- equalities on time scales has received a lot of attention in the literature and has become a major eld in pure and applied mathematics. These dynamic inequalities have a signi cant role in understanding the behavior of solutions of dynamic equations on time scales. The subject of time scale has been created by Stefan Hilger [40] in his Ph.D. Thesis in 1988 for unifying the study of di¤erential and di¤erence equations, and it also extends these classical cases to cases in between, e.g., to the so-called qdi¤erence equations. The general idea is to prove a result for a dynamic inequality where the domain of the unknown function a so-called time scale T, which is nonempty closed subset of the real numbers R, to avoid proving results twice, once in the continuous case which leads to a di¤erential inequality and once again on a discrete case which leads to a di¤erence inequality. The books on the subject of time scale by Bohner and Peterson [22, 23] summarizes and organizes much of vi vii time scale calculus. The recent book of dynamic inequalities on time scales by Agarwal, ORegan and Saker [4, 5] contains most recent basic dynamic inequalities. Also, in recent years, some authors studied the The ; - symmetric quantum calculus, we refer the reader to the papers [24, 25], and the references cited therein. Very recently, some authors have ex- tended classical inequalities by using the ; - symmetric quantum calculus such as Holders inequality, Minkowskis inequality, Dreshers inequality,Cauchys-Schwarzs inequality and their reverses. The authors extended the ; - symmetric quantum calculus and gave de - nitions of the derivatives and integrals. This thesis is devoted to prove some new dynamic inequalities of Hardys type on time scales and a uni ed approach to Copson-Beesack type inequalities on ; - symmetric quantum calculus. Chapter 1. This chapter contains some preliminaries, de nitions and concepts over delta calculus, and ; - symmetric quantum calculus, and basic dynamic inequalities that will be needed in the proofs of the main results. Chapter 2. In this chapter, we present some recent developments of Hardys type inequalities that serve and motivate the contents of this chapter. Next, in the rest of the chapter, we will prove some new generalized weighted dynamic inequalities of Hardys type on a time scale T. The obtained results contain as special cases some published results when the time scale T = R and when T = N. The results, to to the best of the authors knowledge, are essentially new. Chapter 3. In this chapter, we present some recent developments of Copson and Beesack type inequalities that serve and motivate the contents of the next sections. Next, in the rest of the chapter, we will introduce some improvements on ; - symmetric quantum calculus and will use them in proving some new theorems that unify proofs of the Copson inequalities for all values of the exponent k and we will prove that the approach that has been given by Beesack is also valid for the ; - symmetric quantum calculus. |