Schur-Convexity for a Class of Symmetric Functions
Shu-Hong wang1,, Shu-Ping Bai1
1College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China
Abstract | |
1. | Introduction |
2. | Definitions and Lemmas |
3. | Proof of Theorems |
4. | Conclusion |
Acknowledgement | |
Competing Interests | |
References |
Keywords: Symmetric function, Schur- convex function, Schur –concave function, convex function, continuous function, majorized
Received June 11, 2015; Revised February 03, 2016; Accepted February 11, 2016
Copyright © 2016 Science and Education Publishing. All Rights Reserved.Cite this article:
- Shu-Hong wang, Shu-Ping Bai. Schur-Convexity for a Class of Symmetric Functions. Turkish Journal of Analysis and Number Theory. Vol. 4, No. 1, 2016, pp 16-19. http://pubs.sciepub.com/tjant/4/1/3
- wang, Shu-Hong, and Shu-Ping Bai. "Schur-Convexity for a Class of Symmetric Functions." Turkish Journal of Analysis and Number Theory 4.1 (2016): 16-19.
- wang, S. , & Bai, S. (2016). Schur-Convexity for a Class of Symmetric Functions. Turkish Journal of Analysis and Number Theory, 4(1), 16-19.
- wang, Shu-Hong, and Shu-Ping Bai. "Schur-Convexity for a Class of Symmetric Functions." Turkish Journal of Analysis and Number Theory 4, no. 1 (2016): 16-19.
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1. Introduction
Throughout this paper, let be a non-empty open interval in
. A class of functions
are defined as
![]() | (1.1) |
where
If or
, in [1] chu ed al. proved
Theorem A([1, Theorem 1.1]). Suppose that is an open interval and
is a continuous function. The function
![]() | (1.2) |
is Schur-convex (concave) on if and only if
is convex (concave) on
.
Theorem B ([1, Theorem 1.2.]). Suppose that is an open interval and
is a continuous function. The function
![]() | (1.3) |
is Schur-convex (concave) on if and only if
is convex (concave) on
.
If , in [2] I. Franjić and J. Pečarić proved
Theorem C ([2, Theorem 4.]). If , then the following statements are equivalent:
(a) The function
![]() | (1.4) |
is Schur-convex on .
(b) For all , we have
(c) The function is 4-convex on
.
Remark ([2, Remark3.]). In [2], I. Franjić and J. Pečarić have proven that being a convex function is not a sufficient condition for
to be Schur-convex.
Schur-convexity has aroused the interest of many researchers, and numerous papers have been devoted to it. For example, in [3, 4, 6, 7], some related results were given.
The purpose of this paper is to prove the following results:
Theorem 1.1. Let be an open interval, and
be a twice differentiable mapping such that
is integrable. If
and
is convex (concave) on
, then function
is Schur-convex (concave) on
.
Theorem 1.2. Let be an open interval, and
be a twice differentiable mapping such that
is integrable. If
, and the function
is Schur-convex (concave) on
, then
is convex (concave) on
.
2. Definitions and Lemmas
In order to prove our result, we shall need several Definitions and Lemmas, which we present in this section.
Definition 2.1([5]). Let
.
is said to be majorized by
(in symbols
) if
![]() |
and
![]() |
where and
are rearrangements of
and
in a descending order.
Definition 2.2([5]). Let . The function
is said to be a Schur-convex function on
if
on
implies
.
is said to be a Schur-concave function on
if and only if
is Schur-convex.
Lemma 2.1 (Schur-Ostrowski Theorem) ([5]). Let be a symmetric convex set with nonempty interior, and
be a continuous symmetric function on
. If
is differentiable in
. Then
is Schur-convex (concave) on
if and only if
![]() |
for all .
Lemma 2.2. Let be an open interval,
with
. If
and
is a twice differentiable mapping such that
is integrable, then the following identity holds
![]() | (2.1) |
where
![]() |
Proof. It suffices to note that
![]() | (2.2) |
From integrating by part the right-hand sides of (2.2), we can state:
![]() | (2.3) |
and
![]() | (2.4) |
Using (2.3) and (2.4) in (2.2), it follows that
![]() |
Thus, using the change of the variable for
and by multiplying both sides by
, we have the conclusion (2.1).
This completes the proof.
Lemma 2.3. Let be an open interval, and
be a twice differentiable mapping such that
is integrable. If
, and
with
. Then it follows that
![]() |
where .
Proof. From and
, then
![]() | (2.5) |
and
![]() | (2.6) |
So, by Integral mean value theorem, we have
![]() |
where .
This completes the proof.
Lemma 2.4. Suppose that be defined as in (1.1). If
has continuous second order derivatives on
, then
![]() |
for all .
Proof. For any , from (1) together with the L’Hospital’s rule we clearly see that
![]() |
Making use of similar arguments for , and we get
![]() |
This completes the proof.
3. Proof of Theorems
Proof of Theorem 1.1. The proof is divided into three cases.
Case 1. If , then Lemma 2.4 leads to
![]() |
Case 2. If , then (1.1) leads to
![]() | (3.1) |
and
![]() | (3.2) |
By Lemma 2.1 and Lemma 2.2, we have
![]() |
![]() |
If and
is convex, then
Therefore, is Schur convex on
from Lemma 2.1.
Case 3. If , since
is a symmetric function, we also can conclude that
is Schur convex on
from
and the convexity of
.
It follows from the similar arguments as above that is Schur-concave on
if
and
is concave on
, which completes the proof.
Proof of Theorem 1.2. If is Schur-convex on
, then from (1.1) and Definition 1 together with the fact that
we have
![]() |
If by Lemma 2.3, we have
![]() |
where .
Namely , and so, since
and
are arbitrary, we conclude that
is convex.
Since is a symmetric function, if
we also can conclude that
is convex.
It follows from the similar arguments as above that is concave on
if
is Schur- concave on
, which completes the proof.
4. Conclusion
In this paper, we discuss Schur convexity for a class of symmetric functions and obtain the following results:
Theorem 1.1. Let be an open interval, and
be a twice differentiable mapping such that
is integrable. If
and
is convex (concave) on
, then function
is Schur-convex (concave) on
.
Theorem 1.2. Let be an open interval, and
be a twice differentiable mapping such that
is integrable. If
, and the function
is Schur-convex (concave) on
, then
is convex (concave) on
.
Acknowledgement
This work was partially supported by the National Natural Science Foundation of China under Grant No. 11361038 and by the Inner Mongolia Autonomous Region Natural Science Foundation Project under Grant No. 2015MS0123, China.
Competing Interests
The authors declare that they have no competing interests.
References
[1] | Y. Chu, G.Wang, X. Zhang, Schur convexity and Hadamard’s inequality, Math. Inequal. Appl., 13 (4) (2010), 725-731. | ||
![]() | View Article | ||
[2] | I. Franjić, J. Pečarić, Schur-convexity and the Simpson formula, Appl. Math. Lett., (2011). | ||
![]() | |||
[3] | N. Elezović, J. Pečarić, A note on Schur-convex functions, Rocky Mountain J. Math., 30 (3) (2000) , 853-856. | ||
![]() | View Article | ||
[4] | X. Zhang, Y. Chu, Convexity of the integral arithmetic mean of a convex function, Rocky Mountain J. Math. , 40 (3) (2010), 1061-1068. | ||
![]() | View Article | ||
[5] | A. M. Marshall, I. Olkin, B C. Arnold, Inequalities: Theory of Majorization and its Application (Second Edition). Springer New York, (2011). 101. | ||
![]() | View Article PubMed | ||
[6] | H. N. Shi, J. Zang, Schur convexity, Schur geometric and Schur harmonic convexity of dual form of a class symmetric functions. Journal Mathematical & Inequalities, 8(2) (2014), 349-358. | ||
![]() | View Article | ||
[7] | H. N. Shi, J. Zang, Schur-convexity of dual form of some symmetric functions, Journal of Inequalities and Applications, 295 (2013), 9 papes. | ||
![]() | |||