On the two-dimensional Hardy inequality

The two-dimensional Hardy operator is characteried with two conditions. In the case were one of the weightfunctions is of product typeone of the conditions is a pure constant, so the reslut coincide with earlier results.Godkänd; 2009; 20090612 (evan)</p

Weighted inequalities of Hardy-type and their limiting inequalities

This thesis deals with various generalizations of two famous inequalities namely the Hardy inequality and the Pólya-Knopp inequality and the relation between them. 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 the idea of using the weighted Hardy inequality to receive the weighted Pólya-Knopp inequality as a natural limiting inequality is investigated and some problems that arises are discussed. In Chapter 3 a new necessary and sufficient condition for the weighted Hardy inequality is proved and also used to give a new necessary and sufficient condition for a corresponding weighted Pólya-Knopp type inequality. In Chapter 4 a new two-dimensional Pólya-Knopp inequality is proved. This inequality may be regarded as a natural endpoint inequality of the famous two-dimensional Hardy inequality by E. Sawyer, which is characterized by three independent integral conditions while our endpoint inequality is characterized by one condition. In Chapter 5 the three necessary and sufficient conditions for the two- dimensional version of the Hardy inequality given by E. Sawyer are investigated and compared with the corresponding conditions in one dimension. Moreover, the corresponding endpoint problems and conditions are pointed out. In Chapter 5 we also prove a new two-dimensional Hardy inequality, where the weightfunction on the right hand side is of product type. In this case we only need one integral inequality to characterize the inequality and, moreover, by performing the natural limiting process we receive the same result as in Chapter 4. In Chapter 6 we prove criteria for boundedness between weighted Rn spaces of a fairly general multidimensional Hardy-type integral operator with an Oinarov kernel. The integrals are taken over cones in Rn with origin as a vertex. In Chapter 7 the related results are proved for the limiting geometric mean operator with an Oinarov kernel. Finally, in Chapter 8 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. Moreover, we discuss and comment some very new results and put them into this frame. We also include some new proofs and results e.g. a weight characterization of a general weighted Carleman type inequality.Godkänd; 2003; 20061109 (haneit

On the two-dimensional Hardy inequality

The two-dimensional Hardy operator is characteried with two conditions. In the case were one of the weightfunctions is of product typeone of the conditions is a pure constant, so the reslut coincide with earlier results.Godkänd; 2009; 20090612 (evan)</p

Weighted inequalities for the Sawyer two-dimensional Hardy operator and its limiting geometric mean operator

<p/> <p>We consider <inline-formula><graphic file="1029-242X-2005-374053-i1.gif"/></inline-formula> and a corresponding geometric mean operator <inline-formula><graphic file="1029-242X-2005-374053-i2.gif"/></inline-formula>. E. T. Sawyer showed that the Hardy-type inequality <inline-formula><graphic file="1029-242X-2005-374053-i3.gif"/></inline-formula> could be characterized by three independent conditions on the weights. We give a simple proof of the fact that if the weight <inline-formula><graphic file="1029-242X-2005-374053-i4.gif"/></inline-formula> is of product type, then in fact only one condition is needed. Moreover, by using this information and by performing a limiting procedure we can derive a weight characterization of the corresponding two-dimensional P&#243;lya-Knopp inequality with the geometric mean operator <inline-formula><graphic file="1029-242X-2005-374053-i5.gif"/></inline-formula> involved.</p

Some new Hardy type inequalities and their limiting inequalities

Validerad; 2003; 20070117 (evan)</p

CARLEMAN’S INEQUALITY- HISTORY, PROOFS AND SOME NEW GENERALIZATIONS

ABSTRACT. Carleman’s inequality reads a1 + √ a1a2 +... + k √ a1...ak &lt; e (a1 + a2 +....), where ak, k = 1, 2,...., are positive numbers. In this paper we present some simple proofs of and several remarks (e.g. historical) about the inequality and its corresponding continuous version. Moreover, we discuss and comment on some very new results. We also include some new proofs and results e.g. a weight characterization of a general weighted Carleman type inequality for the case 0 &lt; p ≤ q &lt; ∞. We also include some facts about T. Carleman and his work

Scales of weight characterizations for discrete Hardy and Carleman type inequalities

Godkänd; 2004; 20070507 (ysko

Weight characterizations for the discrete Hardy inequality with kernel

A discrete Hardy-type inequality (∑n=1∞(∑k=1ndn,kak)qun)1/q≤C(∑n=1∞anpvn)1/p is considered for a positive "kernel" d={dn,k}, n,k∈ℤ+, and p≤q. For kernels of product type some scales of weight characterizations of the inequality are proved with the corresponding estimates of the best constant C. A sufficient condition for the inequality to hold in the general case is proved and this condition is necessary in special cases. Moreover, some corresponding results for the case when {an}n=1∞ are replaced by the nonincreasing sequences {an*}n=1∞ are proved and discussed in the light of some other recent results of this type.Validerad; 2006; Bibliografisk uppgift: Paper id:: 18030; 20070111 (evan

Programming as a mediator of mathematical thinking : Examples from upper secondary students exploring the definite integral

We report on three episodes from a case study where upper secondary students numerically explore the definite integral in a Python environment. Our research questions concern how code can mediate and support students' mathematical thinking and what kind of sociomathematical norms emerge as students work together to reach a mutual understanding of a correct solution. The main findings of our investigation are as follows. 1) Students can actively use code as a mediator of their mathematical thinking, and code can even serve as a bridge that helps students to develop their mathematical thinking collaboratively. Further, code can help students to perceive mathematical notions as objects with various properties and to communicate about these properties, even in other semiotic systems than the mathematical language. 2) For the participating students, a common norm was that an acceptable solution is a sufficient condition for the correctness of the solution method although students were aware of a problem in their code, yet also other norms emerged. This demonstrates that learning mathematics with programming can have an effect on what kind of sociomathematical norms emerge in classroom.Validerad;2024;Nivå 1;2024-05-16 (signyg);Fulltext license: CC BY</p

Carleman-Knopp type inequalities via Hardy inequalities

Some new Carleman-Knopp type inequalities are proved as "end point" inequalities of modern forms of Hardy's inequalities. Both finite and infinite intervals are considered and both the cases p q and q &lt; p are investigated. The obtained results are compared with similar results in the literature and the sharpness of the constants is discussed for the power weight case. Moreover, some reversed Carleman-Knopp inequalities are derived and applied.Validerad; 2001; 20061020 (evan)</p