الفهرس 
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Abstract
In 1920, Hardy [34] first stated and proved his famous inequality of the form ∑_(n=1)^∞▒(1/n ∑_(k=1)^n▒〖u(k)〗)^p ≤(p/(p-1))^p ∑_(n=1)^∞▒〖u^p (n) 〗,p>1, where u is a sequence of nonnegative numbers. Later, in 1925, he proved the inequality ∫_0^∞▒〖(1/x ∫_0^x▒f(t)dt)^p dx≤(p/(p-1))^p ∫_0^∞▒〖f^p (x)dx,〗〗 where f is nonnegative, f^p is integrable and convergent over (0,1) and p > 1. In 1927, Hardy and Littlewood [35] showed that this integral inequality holds with reversed sign when 0 < p < 1, provided that the integral ∫_0^x▒f(t)dt is replaced by∫_x^∞▒f(t)dt. Since then, these types of inequalities have been studied widely and alot of generalizations and modifications have been established for these types of inequalities. These studies and contributions were established by Copson [23, 24], Knopp [42], Landau [49, 50], Littlewood [51], Pòlya [69, 68], Riesz [71, 72] and Schur [80]. For more information see [1, 36, 47, 48, 67] and references cited therein. We focus our effort on applying the properties of a generalized discrete Hardy operator to prove the discrete analogue of the properties of Muckenhoupt and Gehring classes and establish the self-improving and transition properties between the two classes. Furthermore, we also prove some modified Hardy-type inequalities, which we shall use in studying the structure of the discrete Lorentz space, the relation between different norms defined on these classes and obtaining some Hölder-type inequalities in terms of these different norms. These equivalence relations give some new weighted Hardy- and reverse Hardy-type inequalities for sequences in the classical Lorentz space l^(p,q) and some new weighted Hardy- and reverse Hardy-type inequalities with weights ω in the B_q-class for sequences in the weighted Lorentz space Λ_q (ω). Our main concern in this thesis is to obtain the best discrete results analogous to those in the continuous case. Our work in this thesis is divided into the following folds: Proving some properties of a generalized Hardy operator that are essential in the proofs of the main results; Proving some basic properties of the discrete Muckenhoupt and Gehring classes A_p and G_q; Proving the self-improving properties and some fundamental inclusion and transition relations between the two classes A_p and G_q; Proving some sharp results concerning the monotone weights in the classes A_p and G_q and the weighted classes A_p^λ and G_q^λ; Establishing the exact values of the Muckenhoupt norm and the Gehring norm for the power-low sequences {n^α}; Proving a modified Hardy-type inequality and use in proving some inequalities relating different norms defined on the classical Lorentz sequence space l^(p,q) and the weighted Lorentz sequence space Λ_q (ω); Proving some new weighted Hardy-type inequalities for sequences in the classical Lorentz space l^(p,q) and weighted Lorentz space Λ_q (ω) with ωϵB_q.