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A Strengthened Carleman's Inequality

Yan Ping, Sun Guozheng
1999 Journal of Mathematical Analysis and Applications  
w x In 2 , the 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.  ... 
doi:10.1006/jmaa.1999.6583 fatcat:ko6xxtq2tbgyfgx4vf2ziohbxu

On strengthened weighted Carleman's inequality

Aleksandra Čižmešija, Josip Pecarić, Lars–Erik Persson
2003 Bulletin of the Australian Mathematical Society  
apply this result in obtaining a sharpened version of the weighted 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.  ... 
doi:10.1017/s0004972700037886 fatcat:nflnnu3oi5dylpe4x3mtatwjaa

Carleman's inequality for finite series

N.G. de Bruijn
1963 Indagationes Mathematicae (Proceedings)  
CARLEMAN [2] proved the following 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.  ... 
doi:10.1016/s1385-7258(63)50050-x fatcat:qsm52lyeajdzzlmmtd6mpq4rqi

New Strengthened Carleman's Inequality and Hardy's Inequality

Haiping Liu, Ling Zhu
2007 Journal of Inequalities and Applications  
Introduction The following 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] .  ... 
doi:10.1155/2007/84104 fatcat:7hxtikgstjbydecnp63onxyj4m

A Note on Carleman's Inequality [article]

Peng Gao
2007 arXiv   pre-print
We study a weighted version of 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.  ... 
arXiv:0706.2368v1 fatcat:gthv4odrjjagfni5jv5dx6qkdu

Note on the Carleman's inequality and Hardy's inequality

Lü Zhongxue, Gao Youcai, Wei Yuxiang
2010 Computers and Mathematics with Applications  
In this article, using the properties of power mean and induction, new strengthened 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.  ... 
doi:10.1016/j.camwa.2009.09.001 fatcat:udf7v5dcibaihgs6s5kmmi4imy

Finite Sections of Weighted Carleman's Inequality [article]

Peng Gao
2007 arXiv   pre-print
We study finite sections of weighted 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 .  ... 
arXiv:0707.0077v1 fatcat:5azqvp3xg5gstdhd3e7ufepig4

ON A HARDY-CARLEMAN'S TYPE INEQUALITY

Bicheng Yang
2005 Taiwanese journal of mathematics  
In this paper, we prove that the constant factor in the Hardy-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.  ... 
doi:10.11650/twjm/1500407854 fatcat:rl6a5r2vmzbifmrcvnhj6dhss4

Some refinements and generalizations of Carleman's inequality

Dah-Yan Hwang
2004 International Journal of Mathematics and Mathematical Sciences  
We give some refinements and generalizations of 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.  ... 
doi:10.1155/s0161171204311397 fatcat:a6t7nxu2lfaetpckwk5fba3dum

APPLICATIONS OF TAYLOR SERIES FOR CARLEMAN'S INEQUALITY THROUGH HARDY INEQUALITY

MOHAMMED MUNIRU IDDRISU, CHRISTOPHER ADJEI OKPOTI
2015 Korean Journal of Mathematics  
Also, we prove the 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.  ... 
doi:10.11568/kjm.2015.23.4.655 fatcat:m7wc4eieyvbu3oxaloy4qa4jsa

The Carleman's Inequality for a Negative Power Number

Nguyen Thanh Long, Nguyen Vu Duy Linh
2001 Journal of Mathematical Analysis and Applications  
By the method of indeterminate coefficients we prove the 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].)  ... 
doi:10.1006/jmaa.2000.7422 fatcat:arsuwrys6nfnjp2zzkapokjgvu

Some Inequalities Involving the Constante, and an Application to Carleman's Inequality

Yang Bicheng, Lokenath Debnath
1998 Journal of Mathematical Analysis and Applications  
As an application, we prove a strengthened 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.  ... 
doi:10.1006/jmaa.1997.5617 fatcat:tpkzdzznknbzld4sbaiyxj3f7u

On the coefficients of an expansion of (1+1/x)^x related to Carleman's inequality [article]

Yue Hu, Cristinel Mortici
2014 arXiv   pre-print
In this note, we present new properties for a sequence arising in some refinements of 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  ... 
arXiv:1401.2236v1 fatcat:e4iwpyanuran7j3gzxh4thge5u

Estimates of (1+x)1/x involved in Carleman's inequality and Keller's limit

Cristinel Mortici, X.J. Jang
2015 Filomat  
Debnath in [Some 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.  ... 
doi:10.2298/fil1507535m fatcat:5a35rvpzkfgqlhluyryvmjvp2u

REFINEMENTS OF CARLEMAN'S INEQUALITY

Yu-Ming Chu, Xiao-Ming Zhang, Qian Xu
Poincare Journal of Analysis & Applications   unpublished
In this paper, we prove that the 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 .  ... 
fatcat:b5zgw3fl2rhxnaxyjylqljjfka
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