Hardy and Carleman type inequalities

Abstract: This thesis deals with various generalizations of the inequalities by Carleman, Hardy and Pólya-Knopp. In Chapter 1 we give an introduction and overview of the area that serves as a frame for the rest of the thesis. In Chapter 2 we consider Carleman's inequality, which may be regarded as a discrete version of Pólya-Knopp's inequality and also as a natural limiting inequality of the discrete Hardy inequality. We present several simple proofs of and \ remarks (e.g. historical) about this inequality. In Chapter 3 we give some sharpenings and generalizations of Carleman's inequality. We discuss and comment on these results and put them into the frame presented in the previous chapter. We also include some new proofs and results. In Chapter 4 we give a new weight characterization of the weighted Hardy inequality for decreasing functions. In Chapter 5 we use the results from Chapter 4 to give a new weight characterization of the weighted Pólya-Knopp inequality for decreasing functions and we also give a new scale of weightconditions for the embedding L^{p}(v) to L^{q}(u) for the case 1