Abstract
By the properties of a Schur-convex function, Schur-convexity of the dual form of some symmetric functions is simply proved.
MSC:26D15, 05E05, 26B25.
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1 Introduction
Throughout the article, 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.
For convenience, we introduce some definitions as follows.
Let and .
-
(i)
means for all .
-
(ii)
Let , is said to be increasing if implies . φ is said to be decreasing if and only if −φ is increasing.
Let and .
-
(i)
x is said to be majorized by y (in symbols ) if for and , where and are rearrangements of x and y in a descending order.
-
(ii)
Let , is said to be a Schur-convex function on Ω if on Ω implies . φ is said to be a Schur-concave function on Ω if and only if −φ is Schur-convex function on Ω.
Let and .
-
(i)
is said to be a convex set if , implies .
-
(ii)
Let be a convex set. A function is said to be a convex function on Ω if
for all and all . φ is said to be a concave function on Ω if and only if −φ is a convex function on Ω.
-
(iii)
Let . A function is said to be a log-convex function on Ω if the function lnφ is convex.
Definition 4 [1]
-
(i)
is called a symmetric set, if implies for every permutation matrix P.
-
(ii)
The function is called symmetric if for every permutation matrix P, for all .
Theorem A (Schur-convex function decision theorem [[1], p.84])
Let be symmetric and have a nonempty interior convex set. is the interior of Ω. is continuous on Ω and differentiable in . Then φ is the Schur-convex (Schur-concave) function if and only if φ is symmetric on Ω and
holds for any .
The Schur-convex functions were introduced by Schur in 1923 and have important applications in analytic inequalities, elementary quantum mechanics and quantum information theory. See [1].
In recent years, many scholars use the Schur-convex function decision theorem to determine the Schur-convexity of many symmetric functions.
Xia et al. [3] proved that the symmetric function
is Schur-convex on .
Chu et al. [4] proved that the symmetric function
is Schur-convex on and Schur-concave on .
Xia and Chu [5] proved that the symmetric function
is Schur-convex on and Schur-concave on .
Xia and Chu [6] also proved that the symmetric function
is Schur-convex on .
Mei et al. [7] proved that the symmetric function
is Schur-convex on . More results for Schur convexity of the symmetric functions, we refer the reader to [8].
In this paper, by the properties of a Schur-convex function, we study Schur-convexity of the dual form of the above symmetric functions, and we obtained the following results.
Theorem 1 The symmetric function
is a Schur-concave function on .
Theorem 2 The symmetric function
is a Schur-convex function on .
Theorem 3 The symmetric function
is a Schur-convex function on .
Theorem 4 The symmetric function
is a Schur-convex function on .
Theorem 5 The symmetric function
is a Schur-convex function on .
2 Lemmas
To prove the above three theorems, we need the following lemmas.
If φ is symmetric and convex (concave) on a symmetric convex set Ω, then φ is Schur-convex (Schur-concave) on Ω.
Lemma 2 [[2], p.64]
Let , . Then logφ is Schur-convex (Schur-concave) if and only if φ is Schur-convex (Schur-concave).
Let be an open convex set, . For , define one variable function on the interval . Then φ is convex (concave) on Ω if and only if g is convex (concave) on for all .
Lemma 4 Let and . Then the function is concave on , where
Proof
where
where
Thus,
and then , that is, is concave on .
The proof of Lemma 4 is completed. □
Lemma 5 Let and . Then the function is convex on , where
Proof
where
where
By the Cauchy inequality, we have
From it follows that , hence , and then , that is, is convex on .
The proof of Lemma 5 is completed. □
Lemma 6 Let and . Then the function is convex on , where
Proof
where
where
By the Cauchy inequality, we have
From it follows that , hence , and then , that is, is convex on .
The proof of Lemma 6 is completed. □
Lemma 7 Let and . Then the function is convex on , where
Proof
where
where
By the Cauchy inequality, we have
From it follows that , hence , and then , that is, is convex on .
The proof of Lemma 7 is completed. □
Lemma 8 Let and . Then the function is convex on , where
Proof
where
where
By the Cauchy inequality, we have
Let . From it follows that . Since
so , and then , that is, is convex on .
The proof of Lemma 8 is completed. □
3 Proof of main results
Proof of Theorem 4 For any , by Lemma 3 and Lemma 7, it follows that is convex on . Obviously, is also convex on , and then is convex on . Furthermore, it is clear that is symmetric on . By Lemma 1, it follows that is Schur-convex on , and then from Lemma 2 we conclude that is also Schur-convex on .
The proof of Theorem 4 is completed. □
Similar to the proof of Theorem 4, we can use Lemma 4, Lemma 5, Lemma 6 and Lemma 8 respectively to prove Theorem 1, Theorem 2, Theorem 3 and Theorem 5; therefore we omit the details of the proof.
Remark 1 Using the Schur-convex function decision theorem, Liu et al. [9] have proved Theorem 3. Xia and Chu [10] have proved that the symmetric function
is a Schur-convex function on .
The reader may wish to prove the inequality (12) by the properties of a Schur-convex function.
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Acknowledgements
The work was supported by Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHR (IHLB)) (PHR201108407). Thanks for the help.
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Shi, HN., Zhang, J. Schur-convexity of dual form of some symmetric functions. J Inequal Appl 2013, 295 (2013). https://doi.org/10.1186/1029-242X-2013-295
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DOI: https://doi.org/10.1186/1029-242X-2013-295