Abstract
The judgement theorems of Schur geometric and Schur harmonic convexities for a class of symmetric functions are given. As their application, some analytic inequalities are established.
MSC:26D15, 05E05, 26B25.
Similar content being viewed by others
1 Introduction
Throughout this paper, ℝ denotes the set of real numbers, denotes n-tuple (n-dimensional real vectors), the set of vectors can be written as
In particular, the notations ℝ and denote and , respectively.
Let be a permutation of , all permutations are totally n!. The following conclusion is proved in [[1], pp.127-129].
Theorem A Let be a symmetric convex set, and let φ be a Schur-convex function defined on A with the property that for each fixed , is convex in z on . Then, for any ,
is Schur-convex on
Furthermore, the symmetric function
is also Schur-convex on B.
Theorem A is very effective for judgement of the Schur-convexity of the symmetric functions of the form (2), see the references [1] and [2].
The Schur geometrically convex functions were proposed by Zhang [3] in 2004. Further, the Schur harmonically convex functions were proposed by Chu and Lü [4] in 2009. The theory of majorization was enriched and expanded by using these concepts [5–15]. Regarding Schur geometrically convex functions and Schur harmonically convex functions, the aim of this paper is to establish the following judgement theorems which are similar to Theorem A.
Theorem 1 Let be a symmetric geometrically convex set, and let φ be a Schur geometrically convex (concave) function defined on A with the property that for each fixed , is GA convex (concave) in z on . Then, for any ,
is Schur geometrically convex (concave) on
Furthermore, the symmetric function
is also Schur geometrically convex (concave) on B.
Theorem 2 Let be a symmetric harmonically convex set, and let φ be a Schur harmonically convex (concave) function defined on A with the property that for each fixed , is HA convex (concave) in z on . Then, for any ,
is Schur harmonically convex (concave) on
Furthermore, the symmetric function
is also Schur harmonically convex (concave) on B.
2 Definitions and lemmas
In order to prove some further results, in this section we recall useful definitions and lemmas.
Let and .
-
(i)
We say y majorizes x (x is said to be majorized by y), denoted by , if for and , where and are rearrangements of x and y in a descending order.
-
(ii)
Let , a function is said to be a Schur-convex function on Ω if on Ω implies . A function φ is said to be a Schur-concave function on Ω if and only if −φ is Schur-convex function on Ω.
Let and , . A set is said to be a convex set if implies .
-
(i)
A set is called a symmetric set if implies for every permutation matrix P.
-
(ii)
A function is called symmetric if for every permutation matrix P, for all .
Definition 4 Let , and .
-
(i)
[[3], p.64] A set Ω is called a geometrically convex set if for all and such that .
-
(ii)
[[3], p.107] A function is said to be a Schur geometrically convex function on Ω if on Ω implies . A function φ is said to be a Schur geometrically concave function on Ω if and only if −φ is a Schur geometrically convex function.
Definition 5 [17]
Let .
-
(i)
A set Ω is said to be a harmonically convex set if for every and , where and .
-
(ii)
A function is said to be a Schur harmonically convex function on Ω if implies . A function φ is said to be a Schur harmonically concave function on Ω if and only if −φ is a Schur harmonically convex function.
Definition 6 [18]
Let , be continuous.
-
(i)
A function φ is said to be a GA convex (concave) function on I if
for all .
-
(ii)
A function φ is said to be a HA convex (concave) function on I if
for all .
Lemma 1 [[16], p.57]
Let be a symmetric convex set with a nonempty interior . is continuous on Ω and differentiable on . Then φ is a Schur-convex (Schur-concave) function if and only if φ is symmetric on Ω and
holds for any .
Lemma 2 [[3], p.108]
Let be a symmetric geometrically convex set with a nonempty interior . Let be continuous on Ω and differentiable on . Then φ is a Schur geometrically convex (Schur geometrically concave) function if and only if φ is symmetric on Ω and
holds for any .
Let be a symmetric harmonically convex set with a nonempty interior . Let be continuous on Ω and differentiable on . Then φ is a Schur harmonically convex (Schur harmonically concave) function if and only if φ is symmetric on Ω and
holds for any .
Lemma 4 [18]
Let be an open subinterval, and let be differentiable.
-
(i)
φ is GA-convex (concave) if and only if is increasing (decreasing).
-
(ii)
φ is HA-convex (concave) if and only if is increasing (decreasing).
3 Proofs of main results
Proof of Theorem 1 To verify condition (4) of Lemma 2, denote by the summation over all permutations π such that , . Because φ is symmetric,
Then
Here,
because φ is Schur geometrically convex (concave), and
because is GA convex (concave) in its first argument on . Accordingly, (≤0). This shows that ψ is Schur geometrically convex (concave) on
Notice that
Of course, is Schur geometrically convex (concave) whenever ψ is Schur geometrically convex (concave).
The proof of Theorem 1 is completed. □
Proof of Theorem 2 We only need to verify condition (5) of Lemma 3, the proof is similar to that of Theorem 1 and is omitted. □
Remark 1 In most applications, A has the form for some interval and in this case . Notice that the convexity of φ in its first argument also implies that φ is convex in each argument, the other arguments being fixed, because φ is symmetric.
4 Applications
Let
In 2011, Guan and Guan [20] proved the following theorem through Lemma 2.
Theorem 3 The symmetric function , , is Schur geometrically convex on .
Now, we give a new proof of Theorem 3 by using Theorem 1. Furthermore, we prove the following theorem through Theorem 2.
Theorem 4 The symmetric function , , is Schur harmonically convex on .
Proof of Theorem 3 Let . Then
and
This shows that when , . According to Lemma 2, φ is Schur geometrically convex on . Let , then . From , it follows that . According to Lemma 4(i), φ is GA convex in its single variable on . So is Schur geometrically convex on from Theorem 1. The proof of Theorem 3 is completed. □
Proof of Theorem 4 Let , then
From (7), we get
This shows that when , . According to Lemma 3, φ is Schur harmonically convex on . Let , then . From , it follows that . According to Lemma 4(ii), φ is HA convex in its single variable on . So is Schur harmonically convex on from Theorem 2. The proof of Theorem 4 is completed. □
By using Theorem A, the following conclusion is proved in [[1], p.129].
The symmetric function
is Schur-convex on .
Now we use Theorem 1 and Theorem 2, respectively, to study Schur geometric convexity and Schur harmonic convexity of .
Theorem 5 The symmetric function is Schur geometrically convex and Schur harmonically concave on .
Proof Let , then . Thus,
According to Lemma 2, is Schur geometrically convex on . Let , where , , then . From , it follows that . According to Lemma 4(i), φ is GA convex in its single variable on . So is Schur geometrically convex on from Theorem 1.
It is easy to check that
According to Lemma 3, is Schur harmonically concave on . Let . when . According to Lemma 4(ii), φ is HA concave in its single variable on . So is Schur harmonically concave on from Theorem 2. □
Remark 2 Let
where , . Then
From Theorem 5, it follows that
By using Theorem A, the following conclusion is proved in [[1], p.129].
The symmetric function
is Schur-concave on .
By applying Theorem 2, we further obtain the following result.
Theorem 6 The symmetric function is Schur harmonically convex on .
Proof Let . According to the proof of Theorem 5, is Schur harmonically concave on . Let . From the definition of Schur harmonically convex, it follows that is Schur harmonically convex on . Let , where , . Then . With the fact that for , it follows that φ is HA convex in its single variable on . So, from Theorem 2, is Schur harmonically convex on . □
Remark 3 From Theorem 6 and (10), it follows that
where , .
Remark 4 It needs further discussion that is Schur geometrically convex on .
References
Marshall AW, Olkin I, Arnold BC: Inequalities: Theory of Majorization and Its Application. 2nd edition. Springer, New York; 2011.
Shi H-N: Schur convexity of three symmetric functions. J. Hexi Univ. 2011, 27(2):13–17. (in Chinese)
Zhang XM: Geometrically Convex Functions. An’hui University Press, Hefei; 2004. (in Chinese)
Chu Y-M, Lü Y-P: The Schur harmonic convexity of the Hamy symmetric function and its applications. J. Inequal. Appl. 2009., 2009: Article ID 838529 10.1155/2009/838529
Xia W-F, Chu Y-M: Schur-convexity for a class of symmetric functions and its applications. J. Inequal. Appl. 2009., 2009: Article ID 493759 10.1155/2009/493759
Rovenţa I: Schur convexity of a class of symmetric functions. An. Univ. Craiova, Ser. Mat. Inform. 2010, 37(1):12–18.
Xia W-F, Chu Y-M: On Schur convexity of some symmetric functions. J. Inequal. Appl. 2010., 2010: Article ID 543250 10.1155/2010/543250
Meng J, Chu Y, Tang X: The Schur-harmonic-convexity of dual form of the Hamy symmetric function. Mat. Vesn. 2010, 62(1):37–46.
Chu Y-M, Wang G-D, Zhang X-H: The Schur multiplicative and harmonic convexities of the complete symmetric function. Math. Nachr. 2011, 284(5–6):653–663. 10.1002/mana.200810197
Chu Y-M, Xia W-F, Zhao T-H: Some properties for a class of symmetric functions and applications. J. Math. Inequal. 2011, 5(1):1–11.
Qian W-M: Schur convexity for the ratios of the Hamy and generalized Hamy symmetric functions. J. Inequal. Appl. 2011., 2011: Article ID 131 10.1186/1029-242X-2011-131
Chu Y-M, Xia W-F, Zhang X-H: The Schur concavity, Schur multiplicative and harmonic convexities of the second dual form of the Hamy symmetric function with applications. J. Multivar. Anal. 2012, 105(1):412–421. 10.1016/j.jmva.2011.08.004
Rovenţa I: A note on Schur-concave functions. J. Inequal. Appl. 2012., 2012: Article ID 159 10.1186/1029-242X-2012-159
Xia W-F, Zhan X-H, Wang G-D, Chu Y-M: Some properties for a class of symmetric functions with applications. Indian J. Pure Appl. Math. 2012, 43(3):227–249. 10.1007/s13226-012-0012-5
Shi H-N, Zhang J: Schur-convexity of dual form of some symmetric functions. J. Inequal. Appl. 2013., 2013: Article ID 295 10.1186/1029-242X-2013-295
Wang BY: Foundations of Majorization Inequalities. Beijing Normal University Press, Beijing; 1990. (in Chinese)
Shi H-N: Theory of Majorization and Analytic Inequalities. Harbin Institute of Technology Press, Harbin; 2012. (in Chinese)
Anderson GD, Vamanamurthy MK, Vuorinen M: Generalized convexity and inequalities. J. Math. Anal. Appl. 2007, 335(2):1294–1308. 10.1016/j.jmaa.2007.02.016
Chu Y-M, Sun T-C: The Schur harmonic convexity for a class of symmetric functions. Acta Math. Sci. 2010, 30B(5):1501–1506.
Guan K, Guan R: Some properties of a generalized Hamy symmetric function and its applications. J. Math. Anal. Appl. 2011, 376(2):494–505. 10.1016/j.jmaa.2010.10.014
Acknowledgements
The work was supported by the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHR (IHLB)) (PHR201108407) and the National Natural Science Foundation of China (Grant No. 11101034).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors co-authored this paper together. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Shi, HN., Zhang, J. Some new judgement theorems of Schur geometric and Schur harmonic convexities for a class of symmetric functions. J Inequal Appl 2013, 527 (2013). https://doi.org/10.1186/1029-242X-2013-527
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2013-527