Research Article
Some Refinements and Generalizations of
I. Schur Type Inequalities
Xian-Ming Gu,
1
Ting-Zhu Huang,
1
Wei-Ru Xu,
2
Hou-Biao Li,
1
Liang Li,
1
and Xi-Le Zhao
1
1
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China
2
School of Science, North University of China, Taiyuan 030051, China
Correspondence should be addressed to Ting-Zhu Huang; tingzhuhuang@126.com
Received 18 December 2013; Accepted 6 February 2014; Published 16 March 2014
Academic Editors: L. Acedo, Y.-M. Chu, and B.-N. Guo
Copyright © 2014 Xian-Ming Gu et al. his is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Recently, extensive researches on estimating the value of e have been studied. In this paper, the structural characteristics of I. Schur
type inequalities are exploited to generalize the corresponding inequalities by variable parameter techniques. Some novel upper
and lower bounds for the I. Schur inequality have also been obtained and the upper bounds may be obtained with the help of Maple
and automated proving package (Bottema). Numerical examples are employed to demonstrate the reliability of the approximation
of these new upper and lower bounds, which improve some known results in the recent literature.
1. Introduction
It is well known that
= (1 + (1/))
and
= (1 +
(1/))
+1
are, respectively, monotone increasing and mono-
tone decreasing, and both of them converge to the constant
. In fact, extensive researches for the estimated value of
have been studied [1–4], and the methods for estimating the
value of are of beneit to the improvements of the Hardy
inequality, Carleman inequality, Gamma function inequality,
and so forth [5–13], which is an essential motivation for
this work. Klambauer and Schur have reached the following
conclusion.
Lemma 1 (see [14]). Both
=(1+(1/))
+
and
=(1+
(1/))
+1
(1+(/)) are monotone decreasing sequences if and
only if ≥(1/2).
In fact, it is not hard to prove that
<(1+
1
)
+(1/2)
=
√
1+
1
(1+
1
)
,
(1)
which has been proved by diferent ways; refer to [14–17].
Besides, Fischer and Qi had further studied this issue
(see [18–20]) and they demonstrated that
is a monotone
increasing sequence if and only if
≤
2 ln 3−3 ln 2
2 ln 2− ln 3
=0.409 .... (2)
Moreover, Alzer and Qi have obtained the necessary
and suicient conditions for the monotonicity of generalized
types of
=(1+(/))
+
and
,
()=(1+(/))
+
; see
[21, 22] for details.
Recently, I. Schur has obtained the so-called I. Schur
inequality as follows:
<<
, (3)
where
=
(1+(1/(2+1))) and
=
(1+(1/2)).
It can also solve the problem proposed by Klambauer in
[15]: “Is the contained in a quarter of the interval of (1+
(1/))
<<(1+(1/))
+1
?” herefore, here is included in
the interval length of
=(1+
1
2
)−(1+
1
2+1
)=
1
2(2+1)
≤
1
6
,
=1,2, ....
(4)
Hindawi Publishing Corporation
e Scientific World Journal
Volume 2014, Article ID 709358, 8 pages
http://dx.doi.org/10.1155/2014/709358