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A note on lpnorms of weighted mean matrices
Journal of Inequalities and Applications volume 2012, Article number: 110 (2012)
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.
Let throughout that . For p ≥ 1, let lpbe the Banach space of all complex sequences a = (a n )n ≥ 1with norm
The celebrated Hardy's inequality , Theorem 326] asserts that for p > 1,
Hardy's inequality can be regarded as a special case of the following inequality:
in which C = (cn,k) and the parameter p > 1 are assumed fixed, and the estimate is to hold for all complex sequences a ∈ lp. The lpoperator norm of C is then defined as
It follows that inequality (1.2) holds for any a ∈ lpwhen U1/p≥ ||C||p,pand fails to hold for some a ∈ lpwhen U1/p< ||C||p,p. Hardy's inequality thus asserts that the Cesáro matrix operator C, given by cn,k= 1/n, k ≤ n and 0 otherwise, is bounded on lpand has norm ≤ p/(p - 1). (The norm is in fact p/(p - 1).)
We say a matrix A = (an,k) is a lower triangular matrix if an,k= 0 for n < k and a lower triangular matrix A is a summability matrix if an,k≥ 0 and . We say a summability matrix A is a weighted mean matrix if its entries satisfy:
Hardy's inequality (1.1) now motivates one to determine the lpoperator norm of an arbitrary summability or weighted mean matrix A. Gao  proved the following result:
Theorem 1.1. Let 1 < p < ∞ be fixed. Let A be a weighted mean matrix given by (1.3). If for any integer n ≥ 1, there exists a positive constant 0 < L < p such that
then ||A||p,p≤ p/(p-L).
Theorem 1.2. Let 1 < p < ∞ be fixed. Let A be a weighted mean matrix given by (1.3). If
then ||A||p,p≤ p/(p-L).
The above result of Cartlidge are very handy to use when determining lpnorms of certain weighted mean matrices. We refer the readers to the articles [2, 5–7] for more recent developments in this area.
We note here that by a change of variables in (1.1) and on letting p → +∞, one obtains the following well-known Carleman's inequality , which asserts that for convergent infinite series ∑a n with non-negative terms, one has
with the constant e being best possible.
It is then natural to study the following weighted version of Carleman's inequality:
where the notations are as in (1.3). The task here is to determine the best constant E so that inequality (1.6) holds for any convergent infinite series ∑a n with non-negative terms. Note that Cartlidge's result (Theorem 1.2) implies that when (1.5) is satisfied, then for any a ∈ lp, one has
Similar to our discussions above, by a change of variables in (1.7) and on letting p → +∞, one obtains inequality (1.6) with E = eLas long as (1.5) is satisfied with p replaced by +∞ there.
Note that (1.5) can be regarded as the case p → 1+ of (1.4) while the case p → +∞ of (1.4) suggests the following result:
Corollary 1.1. Suppose that
then inequality (1.6) holds with E = eM.
In fact, the above corollary is a consequence of the following nice result of Bennett (see the proof of [5, Theorem 13]):
Theorem 1.3. Inequality (1.6) holds with
It is shown in the proof of Theorem 13 in  that Corollary 1.1 follows from the above theorem. It is also easy to see that M ≤ L for L defined by (1.5) so that Corollary 1.1 provides a better result than what one can infer from Cartlidge's result as discussed above.
Note that the bound given in Theorem 1.3 is global in the sense that it involves all the λ n 's and it implies the local version Corollary 1.1, in which only the terms Λ n /λ n and Λn+1/λn+1are involved. It is then natural to ask whether one can obtain a similar result for the lpnorms for p > 1 so that it implies the local version Theorem 1.1. It is our goal in this note to present one such result and as our result is motivated by the result of Bennett, we will first study the limiting case p → +∞, namely weighted Carleman's inequalities in the next section before we move on to the lpcases in Section 3.
2. A discussion on weighted Carleman's inequalities
In this section, we study the weighted Carleman's inequalities. Our goal is to give a different proof of Theorem 1.3 than that given in  and discuss some variations of it. It suffices to consider the cases of (1.6) with the infinite summations replaced by any finite summations, say from 1 to N ≥ 1 here. Our starting point is the following result of Pečarić and Stolarsky [, (2.4)], which is an outgrowth of Redheffer's approach in :
where b is any positive sequence and
We now discard the last term on the left-hand side of (2.1) and make a change of variables to recast inequality (2.1) as
Now, a further change of variables b n ↦ λn+1b n /λ n allows us to recast the above inequality as
If one now chooses b n = (Λn+1/λn+1)/(Λ n /λ n ) such that Λ n (b n /λ n - 1/λn+1) = 1, then one gets immediately the following:
from which the assertion of Theorem 1.3 can be readily deduced.
Another natural choice for the values of b n 's is to set . From this we see that and substituting these values for b n 's we obtain via (2.2):
One checks easily that the above inequality implies Corollary 1.1.
We now consider a third choice for the b n 's by setting and it follows from this and (2.2) that
from which we deduce the following
Corollary 2.1. Suppose that
then inequality (1.6) holds with E = eM.
We point out here that the above corollary also provides a better result than what one can infer from Cartlidge's result and it also follows from Theorem 1.3.
We note here that the optimal choice for the b n 's will be to choose them to satisfy
In general, it is difficult to solve for the b n 's from the above equations. But we can solve b1 to get b1 = eLλ1/((eL- 1)λ2) and if we set b n = (Λn+1/λn+1)/(Λ n /λ n ) for n ≥ 2, we can then deduce from (2.2) that
Note that this gives an improvement upon Theorem 1.3 as long as λ2/Λ2 > e-L. Similarly, one obtains
3. The lpcases
We now return to the discussions on the general lpcases. Again it suffices to consider the cases of (1.7) with the infinite summations replaced by any finite summations, say from 1 to N ≥ 1 here. We may also assume that a n ≥ 0 for all n. With the discussions of the previous section in mind, here we seek for an lpversion of (2.2). Fortunately, this is available by noting that it follows from inequality (4.3) of  that
where w n 's are positive parameters and
By a change of variables , we can recast inequality (3.1) as
With another change of variables, w n /wn+1↦ b n , we can further recast the above inequality as
Note that if one makes a change of variables , then inequality (2.2) follows from the above inequality upon letting p → +∞.
One can then deduce Theorem 1.1 by choosing b n = (1-L λ n /(p Λ n ))-(p-1)in (3.2) (see ). It is easy to see that this gives back inequality (2.3) upon letting p → +∞ by a change of variables and setting L = M. We note here that the b n 's are so chosen so that the following relations are satisfied:
Now, to get the lpanalogues of Theorem 1.3, we just need to note that in the p → +∞ case, one obtains Corollary 1.1 by setting and the conclusion of Corollary 1.1 follows by requiring that Λ n (b n /λ n - 1/λn+1) ≥ 1 for the so chosen b n 's. If one instead chooses the b n 's so that the conditions Λ n (b n /λ n - 1/λn+1) = 1 are satisfied, then Theorem 1.3 will follow. Now, in the lpcases, the choice of the b n 's so that the conditions (3.3) are satisfied implies Theorem 1.1 as (1.4) implies that for the so chosen b n 's,
Thus, in order to obtain an result analogue to Theorem 1.3 for the lpcases, we are then motivated to take the b n 's so that the following conditions are satisfied:
We then easily deduce the following lpanalogue of Theorem 1.3:
Theorem 3.1. Let 1 < p < ∞ be fixed. Let A be a weighted mean matrix given by (1.3). If for any integer n ≥ 1, there exists a positive constant 0 < L < p such that
then ||A||p,p≤ p/(p-L).
It is easy to see by induction that Theorem 3.1 implies Theorem 1.1. Of course one should really choose the b n 's so that the following relations are satisfied:
In general it is difficult to determine the b n 's this way but one can certainly solve for b1 and by choosing other b n 's so that (3.4) are satisfied, one can obtain a slightly better result than Theorem 3.1, we shall leave the details to the reader.
We note here that the choice of the b n 's satisfying (3.4) corresponds to the following choice for the a n 's in Section 4 of  (these a n 's are not to be confused with the a n 's used in the rest of the article):
We also note here that in the case of λ n = L = 1, on choosing b n 's to satisfy (3.4), we obtain via (3.2) that
This gives an improvement of Hardy's inequality (1.1).
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The author declares that he has no competing interests.