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Mixed means and inequalities of Hardy and Carleman type. (English) Zbl 0947.26021

Abstract of thesis: In this thesis mixed means related to discrete and integral power means of arbitrary real order are considered and then applied in obtaining new proofs of the classical discrete and integral Hardy and Carleman inequality. Moreover, two different multivariable analogues of these integral results are given: a generalization to \(n\)-dimensional balls and to \(n\)-dimensional cells.
After the introduction, the thesis is divided into four chapters. In the first chapter the discrete, as also one-dimensional and multidimensional integral power means and the related mixed \((r,s)\)-means are defined, their basic properties are given and important inequalities that describe the relation between \((r,s)\) and \((s,r)\)-mean are derived.
In the second chapter these relations are used in obtaining Hardy type inequalities. At first, new proofs of the discrete and one-dimensional integral Hardy inequality are presented and after that two \(n\)-dimensional integral versions of that result are formulated and then proved by using proper mixed means from the previous chapter. The derived inequalities are related to the cells and the balls in \(\mathbb{R}^n\). The best possible constants for all inequalities are obtained.
The same method is applied also in the third chapter, where the mixed means are used in proving the discrete and integral Carleman inequality and its two integral generalizations to the \(n\)-dimensional balls and cells. These inequalities are called Levin-Cochran-Lee inequalities. Like in the second chapter, the best possible constants for the obtained relations are considered.
At the end of the thesis, another approach to the inequalities of Levin-Cochran-Lee type is given by recalling one more general integral inequality from 1938, due to Levin. This result is generalized in the fourth chapter. It is shown that the relations from the previous chapter are its special cases. The sharpness of the results is discussed.

MSC:

26D15 Inequalities for sums, series and integrals
26E60 Means
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