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Some New Contributions in the Theory of Hardy Type InequalitiesYimer, Markos Fisseha January 2023 (has links)
In this thesis we derive various generalizations and refinements of some classical inequalities in different function spaces. We consider some of the most important inequalities namely the Hardy, Pólya-Knopp, Jensen, Minkowski and Beckenbach-Dresher inequalities. The main focus is put on the Hardy and their limit Pólya-Knopp inequalities. Indeed, we derive such inequalities even in a general Banach functionsetting. The thesis consists of three papers (A, B and C) and an introduction, which put these papers into a more general frame. This introduction has also independent interest since it shortly describe the dramatic more than 100 years of development of Hardy-type inequalities. It contains both well-known and very new ideas and results. In paper A we prove and discuss some new Hardy-type inequalities in Banach function space settings. In particular, such a result is proved and applied for a new general Hardy operator, which is introduced in this paper (this operator generalizes the usualHardy kernel operator). These results generalize and unify several classical Hardy-type inequalities. In paper B we prove some new refined Hardy-type inequalities again in Banach function space settings. The used (super quadraticity) technique is also illustrated by making refinements of some generalized forms of the Jensen, Minkowski and Beckenbach-Dresher inequalities. These results both generalize and unify several results of this type. In paper C for the case 0<p≤q<∞ we prove some new Pólya-Knopp inequalities in two and higher dimensions with good two-sided estimates of the sharp constants. By using this result and complementary ideas it is also proved a new multidimensional weighted Pólya-Knopp inequality with sharp constant. / In this thesis we derive various generalizations and refinements of some classical inequalities in different function spaces. We consider some of the most important inequalities namely the Hardy, Pólya-Knopp, Jensen, Minkowski and Beckenbach-Dresher inequalities. The main focus is put on the Hardy and their limit, Pólya-Knopp inequalities. Indeed, we derive such inequalities even in a general Banach function setting. We prove and discuss some new Hardy-type inequalities in Banach function space settings. In particular, such a result is proved and applied for a new general Hardy operator. These results generalize and unify several classical Hardy-type inequalities. Next, we prove some new refined Hardy-type inequalities again in Banach function space settings. We used superquadraticity technique to prove refinements of some classical inequalities. Finally, for the case 0<p≤q<∞, we prove some new Pólya-Knopp inequalities in two and higher dimensions with good two-sided estimates of the sharp constants. By using this result and complementary ideas it is also proved a new multidimensional weighted Pólya-Knopp inequality with sharp constant.
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