Abstract
We present some results concerning the lpnorms of weighted mean matrices. These results can be regarded as analogues to a result of Bennett concerning weighted Carleman's inequalities.2000 Mathematics Subject Classification: Primary 47A30.
Highlights
Let throughout that p = 0, 1 + 1 = 1
Similar to our discussions above, by a change of variables ak → a1k/p in (1.7) and on letting p ® +∞, one obtains inequality (1.6) with E = eL as long as (1.5) is satisfied with p replaced by +∞ there
It is natural to ask whether one can obtain a similar result for the lp norms for p > 1 so that it implies the local version Theorem 1.1
Summary
Hardy’s inequality (1.1) motivates one to determine the lp operator norm of an arbitrary summability or weighted mean matrix A. We note here that by a change of variables ak → a1k/p in (1.1) and on letting p ® +∞, one obtains the following well-known Carleman’s inequality [8], which asserts that for convergent infinite series ∑an with non-negative terms, one has an, n=1 k=1 n=1 with the constant e being best possible.
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