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A Strengthened Carleman's Inequality

1999
*
Journal of Mathematical Analysis and Applications
*

w x In 2 , the

doi:10.1006/jmaa.1999.6583
fatcat:ko6xxtq2tbgyfgx4vf2ziohbxu
*Carleman's**inequality*was generalized. In this note, the w x results given in 2 can be further generalized and a new much simpler proof can be given. Ž w x . ... The following*Carleman's**inequality*is well known see 1, Chapt. 9.12 . THEOREM A. Let a G 0, n s 1, 2, . . . , and 0 -Ý ϱ a -ϱ. Then In this note, we shall prove the following theorem. ...##
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On strengthened weighted Carleman's inequality

2003
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Bulletin of the Australian Mathematical Society
*

apply this result in obtaining a sharpened version of the weighted

doi:10.1017/s0004972700037886
fatcat:nflnnu3oi5dylpe4x3mtatwjaa
*Carleman's**inequality*. ... ON STRENGTHENED WEIGHTED*CARLEMAN'S**INEQUALITY*ALEKSANDRA CIZMESIJA, JOSIP PECARIC AND LARS-ERIK PERSSON In this paper we prove a new refinement of the weighted arithmetic-geometric mean*inequality*and ... The study of*Carleman's**inequality*is also covered by a rich literature. ...##
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Carleman's inequality for finite series

1963
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Indagationes Mathematicae (Proceedings)
*

CARLEMAN [2] proved the following

doi:10.1016/s1385-7258(63)50050-x
fatcat:qsm52lyeajdzzlmmtd6mpq4rqi
*inequality*for 00 convergent infinite series ! an with positive terms: 1 00 00 (1.1) ! (a1 ... a.) 1 1• < e ! a •. ...*Carleman's**inequality*, however, is essentially different in nature, and our method for proving (1.3) will have nothing in common with the linear algebra methods used for the proofs of (1.4) and (1.5). ... A much simpler proof of*Carleman's**inequality*(in the strong form (1.1)) was given later by P6LYA (see [3] ), but that method does not throw much light on our problem about the finite series. 3. ...##
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New Strengthened Carleman's Inequality and Hardy's Inequality

2007
*
Journal of Inequalities and Applications
*

Introduction The following

doi:10.1155/2007/84104
fatcat:7hxtikgstjbydecnp63onxyj4m
*Carleman's**inequality*and Hardy's*inequality*are well known. Theorem 1.1 (see [1, Theorem 334] ). ... (1.2) In [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] , some refined work on*Carleman's**inequality*and Hardy's*inequality*had been gained. It is observing that in [3] . ...##
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A Note on Carleman's Inequality
[article]

2007
*
arXiv
*
pre-print

We study a weighted version of

arXiv:0706.2368v1
fatcat:gthv4odrjjagfni5jv5dx6qkdu
*Carleman's**inequality*via*Carleman's*original approach. As an application of our result, we prove a conjecture of Bennett. ... Introduction The well-known*Carleman's**inequality*asserts that for convergent infinite series a n with nonnegative terms, one has There is a rich literature on many different proofs of*Carleman's**inequality*... We shall refer the readers to the survey articles [7] and [5] as well as the references therein for an account of*Carleman's**inequality*. ...##
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Note on the Carleman's inequality and Hardy's inequality

2010
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Computers and Mathematics with Applications
*

In this article, using the properties of power mean and induction, new strengthened

doi:10.1016/j.camwa.2009.09.001
fatcat:udf7v5dcibaihgs6s5kmmi4imy
*Carleman's**inequality*and Hardy's*inequality*are obtained. We also give an answer to the conjectures proposed by X. ... Setting λ n = 1 in Theorem 3.1, then we also get an extension of the strengthened*Carleman's**inequality*as the following. Corollary 3.3. ... Setting λ n = 1 in Theorem 3.4, then we also get an extension of the strengthened*Carleman's**inequality*as the following. Corollary 3.6. ...##
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Finite Sections of Weighted Carleman's Inequality
[article]

2007
*
arXiv
*
pre-print

We study finite sections of weighted

arXiv:0707.0077v1
fatcat:5azqvp3xg5gstdhd3e7ufepig4
*Carleman's**inequality*following the approach of De Bruijn. Similar to the unweighted case, we obtain an asymptotic expression for the optimal constant. ... Introduction The well-known*Carleman's**inequality*asserts that for convergent infinite series a n with nonnegative terms, one has There is a rich literature on many different proofs of*Carleman's**inequality*... Using*Carleman's*original approach in [2] , the author [4] proved the following: In this paper, we consider finite sections of weighted*Carleman's**inequality*(1.1): (1.3) N n=1 G n ≤ µ N N n=1 a n . ...##
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ON A HARDY-CARLEMAN'S TYPE INEQUALITY

2005
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Taiwanese journal of mathematics
*

In this paper, we prove that the constant factor in the Hardy-

doi:10.11650/twjm/1500407854
fatcat:rl6a5r2vmzbifmrcvnhj6dhss4
*Carleman's*type*inequality*is the best possible. A related integral*inequality*with a best constant factor is considered. ... (b) We still can't show that the constant factor in (1.10) is the best possible or not, even if we find that for r > 1 the related integral*inequality*of (1.10) is still (3.2). ... We call (1.9) and (1.10) the Hardy-*Carleman's*type*inequalities*. The main objective of this paper is to prove that the constant factor in (1.9) is the best possible. ...##
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Some refinements and generalizations of Carleman's inequality

2004
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International Journal of Mathematics and Mathematical Sciences
*

We give some refinements and generalizations of

doi:10.1155/s0161171204311397
fatcat:a6t7nxu2lfaetpckwk5fba3dum
*Carleman's**inequality*with weaker condition for weight coefficient. ... (1.2) In [19] , Yuan obtained the refined*Carleman's**inequality*as follows. ... The following*Carleman's**inequality*(see [6, Theorem 334] ) is well known, unless (a n ) is null: ∞ n=1 a 1 a 2 ···a n 1/n < e ∞ n=1 a n . (1.1) The constant is the best possible. ...##
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APPLICATIONS OF TAYLOR SERIES FOR CARLEMAN'S INEQUALITY THROUGH HARDY INEQUALITY

2015
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Korean Journal of Mathematics
*

Also, we prove the

doi:10.11568/kjm.2015.23.4.655
fatcat:m7wc4eieyvbu3oxaloy4qa4jsa
*Carleman's**inequality*through limiting the discrete Hardy*inequality*with applications of Taylor series. ... In this paper, we prove the discrete Hardy*inequality*through the continuous case for decreasing functions using elementary properties of calculus. ... We also established the well known*Carleman's**inequality*through limiting Discrete Hardy*inequality*with applications of Taylor series. ...##
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The Carleman's Inequality for a Negative Power Number

2001
*
Journal of Mathematical Analysis and Applications
*

By the method of indeterminate coefficients we prove the

doi:10.1006/jmaa.2000.7422
fatcat:arsuwrys6nfnjp2zzkapokjgvu
*inequality*∞ n=1 ∞ n=1 a n < ∞. ... Using again Lemma 3(ii) we also obtain the same*inequality*(34) from (33). ... where a n ≥ 0, n = 1 2 INTRODUCTION The following Carleman*inequality*is well known. (See [1, Chap. 9 .12].) ...##
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Some Inequalities Involving the Constante, and an Application to Carleman's Inequality

1998
*
Journal of Mathematical Analysis and Applications
*

As an application, we prove a strengthened

doi:10.1006/jmaa.1997.5617
fatcat:tpkzdzznknbzld4sbaiyxj3f7u
*Carleman's**inequality*. LEMMAS AND A THEOREM 1 LEMMA 2.1. ... Ž .*inequalities*2.6 are true, and so are*inequalities*2.7 . This completes the proof. e y -, n s 1, 2, . . . . ... A STRENGTHENED*CARLEMAN'S**INEQUALITY*Ž w x. ...##
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On the coefficients of an expansion of (1+1/x)^x related to Carleman's inequality
[article]

2014
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arXiv
*
pre-print

In this note, we present new properties for a sequence arising in some refinements of

arXiv:1401.2236v1
fatcat:e4iwpyanuran7j3gzxh4thge5u
*Carleman's**inequality*. ... Introduction The following Carleman*inequality*[3] ∞ n=1 (a 1 a 2 · · · a n ) 1/n < e ∞ n=1 a n , whenever a n ≥ 0, n = 1, 2, 3, . . . , with 0 < ∞ n=1 a n < ∞, has attracted the attention of many authors ...##
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Estimates of (1+x)1/x involved in Carleman's inequality and Keller's limit

2015
*
Filomat
*

Debnath in [Some

doi:10.2298/fil1507535m
fatcat:5a35rvpzkfgqlhluyryvmjvp2u
*inequalities*involving the constant e and an application to*Carleman's**inequality*, J. Math. Anal. Appl., 223 (1998), 347-353]. ... As example, we refer to*inequality*(1) which is the main tool for improving*Carleman's**inequality*in [4] . ... of*Carleman's*type. ...##
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REFINEMENTS OF CARLEMAN'S INEQUALITY

*
Poincare Journal of Analysis & Applications
*
unpublished

In this paper, we prove that the

fatcat:b5zgw3fl2rhxnaxyjylqljjfka
*inequalities*∞ ∑ n=1 (n ∏ k=1 a k) 1 n ≤ e ∞ ∑ n=1 (1 − a n + 13a)a n and ∞ ∑ n=1 [(1 + c n + b)(n ∏ k=1 a k) 1 n ] ≤ e ∞ ∑ n=1 a n hold if a n ≥ 0 (n = 1, 2, · · ·) with ... REFINEMENTS OF*CARLEMAN'S**INEQUALITY*Abstract. ... Introduction Let a n ≥ 0 (n = 1, 2, · · · ) with 0 < ∞ ∑ n=1 a n < +∞, then the well-known*Carleman's**inequality*[1] is given by ∞ ∑ n=1 ( n ∏ k=1 a k ) 1 n < e ∞ ∑ n=1 a n . ...
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