الفهرس 
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Abstract
Historically, mathematical analysis has been the major and significant branch of math- ematics for the last three centuries. Indeed, inequalities became the heart of mathematical analysis. Many great mathematicians have made significant contributions to many new developments of the subject, which led to the discovery of many new inequalities with proofs and useful applications in many fields of mathematical physics, pure and applied mathematics. 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. Pólya [44], which was first published treatise in 1934. In recent years, the study of dynamic inequalities on time scales has received a lot of attention in the literature and has become a major field in pure and applied mathematics. These dynamic inequalities have a significant role in understanding the behavior of solutions of dynamic inequalities on time scales. The subject of time scale has been created by Stefan Hilger [49] in his Ph.D. thesis in 1988 for unifying the study of differential and difference equations, and it also extends these classical cases to cases ”in between”, e.g., to the so-called q—difference equations. The general idea is proving a result for a dynamic inequality where the domain of the unknown function so- called time scale A, which is a nonempty closed subset of the real numbers R to avoid proving results twice once in the continuous case which leads to a differential inequality and once again on a discrete case which leads to a difference inequality. The books by Bohner and Peterson [27, 28] are summarized and organized much of time scale calculus. The recent book of
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dynamic inequalities on time scales by Agarwal, O’Regan and Saker [7, 8] contains most recent basic dynamic inequalities. Also, in recent years, some authors studied the fractional inequalities by using fractional Caputo and Riemann-Liouville derivatives, we refer the reader to the papers [26, 52, 111] and the references cited therein. Very recently, some authors have extended classical in- equalities by using conformable fractional calculus such as Opial’s inequality [100, 101], Hermite-Hadamard’s inequality [32, 56, 104],
Chebyshev’s inequality [12] and Stefensen’s inequality [102]. In [3, 57], the authors extended the calculus of fractional order to con- formable calculus and gave new definitions of the derivatives and integrals. In [22, 72], the authors combined a conformable fractional calculus and a time scale calculus and obtained new fractional calculus on time scales. This thesis is devoted to prove some new dynamic inequalities of Hilbert type, Hardy type and Hölder type on time scales and prove some new dynamic inequalities of reverse of Copson type and Bennett-Leindler type via conformable fractional calculus on time scales. This thesis consists of five chapters and is organized as follows:
Chapter 1 is an introductory chapter and contains some preliminaries, definitions and concepts over delta calculus, nabla calculus, diamond- calculus, delta conformable fractional calculus and nabla conformable fractional calculus on a time scale A.
Chapter 2, is devoted to Hilbert-type inequalities and their extensions on an arbitrary time scale A that serve and motivate the contents of this chapter. Next, in the rest of the chapter, we prove some new forms for Hilbert type inequalities. Also, we state and prove new inequalities of Hilbert type for two variables on time scales and we get new inequalities of Hilbert type for nabla calculus on time scales.
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Chapter 3, in this chapter, we present some recent developments of Hardy type inequalities that serve and motivate the contents of this chapter. Next, in the rest of the chapter, we present dynamic inequalities for Hardy type in quotients with general kernels and measures. Also, we get inequalities of Hardy type via superquadratic functions with general kernels and measures for several variables on time scales and we get dynamic Hardy type inequalities with non-conjugate parameters.
Chapter 4, in this chapter, we investigate some new generalizations and refine- ments for Hölder’s inequality and it’s reverse on time scales through the diamond-α dynamic integral, which is defined as a linear combination of the delta and nabla integrals.
Chapter 5, in this chapter, we get new inequalities for reverse of Copson type for delta conformable fractional calculus on time scales. Also, we get new inequalities for Bennett-Leindler type for nabla conformable fractional calculus on time scales.