الفهرس 
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Abstract
In 1972, Muckenhoupt introduced the class of weights in connection with boundedness properties of the Hardy-Littlewood maximal operator on weighted Lebesegue spaces. Muckenhoupt weights are important tools in harmonic analysis, partial differential equations and quasi-conformal mappings. An important and very useful of property of this weight is the ”self-improving property” which describes the relation between and their classes. In 1973, Gehring extended the Muckenhoupt results and establish a new class of weights that satisfying a reverse Hölder inequality. Gehring class was signled out in connection with local integrability properties of the gradient of quasi-conformal mappings. In 1974, Coifman and Fefferman studied the relation between Muckenhoupt and Gehring classes and connected between these classes. Our aim of this thesis, is to investigate some further properties of the generalized Muckenhoupt class and Gehring class with weights. The main purpose of this work is four folds: 1) To introduce some inclusion properties of generalized Muckenhoupt classes via a generalized Hardy operator 2) To introduce some inclusion properties of generalized Gehring classes via a generalized Hardy operator 3) To introduce some inclusion properties of generalized that connected and via a generalized Hardy operator 4) To introduce some inclusion properties of generalized of -space via a generalized Hardy operator The thesis is divided into three chapters and organized as follows: Chapter 1. In this chapter, which is an introductory chapter, we presented the brief history of Hardy type inequalities in both integral and discrete forms with some generalizations via convexity such as Copson’s type inequalities, Knopp’s type inequalities and Levinson’s type inequalities. Chapter 2. In this chapter, we concerned with some basic definitions, properties of Muckenhoupt classes, Gehring classes and the relations between these classes. Moreover, we will study the ”self-improving” and ”transition” properties of and classes. We conclude this chapter by introducing some relations between and classes via the Hardy operator that serve and motivate the contents of this thesis. Chapter 3. In this chapter, we will prove some fundamental properties of the power mean operator of order which is defined by where are nonnegative functions and Then by employing these properties.