الفهرس 
1 IntroductionOnly 14 pages are availabe for public view
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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
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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.