Abstract
In this paper, we present the Schur convexity and monotonicity properties for the ratios of the Hamy and generalized Hamy symmetric functions and establish some analytic inequalities. The achieved results is inspired by the paper of Hara et al. [J. Inequal. Appl. 2, 387-395, (1998)], and the methods from Guan [Math. Inequal. Appl. 9, 797-805, (2006)]. The inequalities we obtained improve the existing corresponding results and, in some sense, are optimal.
2010 Mathematics Subject Classification: Primary 05E05; Secondary 26D20.
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1 Introduction
Throughout this paper, we denote . For , the Hamy symmetric function [1] is defined as
where r is an integer and 1 ≤ r ≤ n.
The generalized Hamy symmetric function was introduced by Guan [2] as follows
where r is a positive integer.
In [2], Guan proved that both F n (x,r) and are Schur concave in . The main of this paper is to investigate the Schur convexity for the functions and and establish some analytic inequalities by use of the theory of majorization.
For convenience of readers, we recall some definitions as follows, which can be found in many references, such as [3].
Definition 1.1. The n-tuple x is said to be majorized by the n-tuple y (in symbols x ≺ y), if
where 1 ≤ k ≤ n - 1, and x[i]denotes the i th largest component of x.
Definition 1.2. Let E ⊆ ℝnbe a set. A real-valued function F : E → ℝ is said to be Schur convex on E if F(x) ≤ F(y) for each pair of n-tuples x = (x1,..., x n ) and y = (y1,..., y n ) in E, such that x ≺ y. F is said to be Schur concave if -F is Schur convex.
The theory of Schur convexity is one of the most important theories in the fields of inequalities. It can be used in combinatorial optimization [4], isoperimetric problems for polytopes [5], theory of statistical experiments [6], graphs and matrices [7], gamma functions [8], reliability and availability [9], optimal designs [10] and other related fields.
Our aim in what follows is to prove the following results.
Theorem 1.1. Let is an integer, then the function is Schur concave in and increasing with respect to x i (i= 1,2, ...,n).
Theorem 1.2. Let is an integer, then the function is Schur concave in and increasing with respect to x i (i= 1,2,... n).
Corollary 1.1. If and that c ≥ s, then
and
where are the arithmetic and geo-metric means of x, respectively.
Corollary 1.2. If and that c ≥ s, then
and
2 Lemmas
In order to establish our main results, we need several lemmas, which we present in this section.
Lemma 2.1 (see [3]). Let E ⊆ ℝnbe a symmetric convex set with nonempty interior intE and φ : E → ℝ be a continuous symmetric function. If φ is differentiable on intE, then φ is Schur convex (or Schur concave, respectively) on E if and only if
for all i,j = 1,2,...,n and x = (x1,...,x n ) ∈ intE.
The r th elementary symmetric function (see [11]) is defined as
where 1 ≤ r ≤ n is a positive integer, and E n (x, 0) = 1.
By (2.1) and simple computations, we have the following lemma.
Lemma 2.2. Let , if
Then,
Lemma 2.3 (see [11]). Let is an integer and 1 ≤ r ≤ n - 1.
Then,
Another important symmetric function is the complete symmetric function (see [3]), which is defined by
where i1, i2,..., i n are non-negative integer, r ∈ {1, 2,...} and C0(x) = 1.
Lemma 2.4 (see [12]). Let x i > 0, i = 1, 2,..., n, and .
Then,
Lemma 2.5 (see [13]). If , then
Lemma 2.6 (see [14]). If and c ≥ s, then
-
(1)
,
(2).
3 Proof of Theorems
Proof of Theorem 1.1. It is obvious that ϕ r (x) is symmetric and has continuous partial derivatives in . By Lemma 2.1, we only need to prove that
For any fixed 2 ≤ r ≤ n, let and , we have
Differentiating ϕ r (x) with respect to x1 yields
Using Lemma 2.2 repeatedly, we get
Equations (3.2) and (3.3) lead to
where
and
Similarly, we can deduce that
From (3.4) and (3.5), one has
It follows from (3.3) and Lemma 2.3 that
Similarly, we can get B > 0.
It follows from the function is decreasing in (0, +∞) that
Therefore, inequality (3.1) follows from (3.6) and (3.7) together with A > 0 and B > 0.
Next, we prove that is increasing with respect to x i (i= 1,2,...,n).
By the symmetry of ϕ r (x) with respect to x i (i = 1, 2,..., n), we only need to prove that
which can be derived directly from A > 0 and B > 0 together with Equation (3.4).
Proof of Theorem 1.2. It is obvious that is symmetric and has continuous partial derivatives in . By Lemma 2.1, we only need to prove that
For any fixed 2 ≤ r ≤ n, let and . Then,
Differentiating with respect to x1, we have
It follows from Lemma 2.4 that
Equations (3.10) and (3.11) lead to
Similarly, we have
From (3.12) and (3.13), one has
By Lemma 2.5, we know that
The monotonicity of the function in (0, +∞) leads to the conclusion that
Therefore, inequality (3.8) follows from (3.14)-(3.16).
Next, we prove that is increasing with respect to x i (i= 1,2,...,n).
From (3.12) and (3.15), we clearly see that
Inequality (3.17) implies that is increasing with respect to x1, then from the symmetry of with respect to x i (i = 1, 2,..., n) we know that is increasing with respect to each x i (i = 1, 2,..., n).
Proof of Corollary 1.1. By Theorem 1.1 and Lemma 2.6, we have and which imply Corollary 1.1.
Remark 1. Let , then
where (1 - x) = (1 - x1, 1 - x2,... , 1 - x n ), commonly referred to as Ky Fan inequality (see [15]), which has attracted the attention of a considerable number of mathematicians (see [16–20]).
Letting and taking c = 1 in Corollary 1.1, we get
It is obvious that inequality (3.19) can be called Ky Fan-type inequality.
Remark 2. Let x i > 0, i = 1, 2,..., n, the following inequalities
and
are the well-known Weierstrass inequalities (see [11]).
Taking c = s = 1 in Corollary 1.1, one has
and
It is obvious that our inequalities can be called Weierstrass-type inequalities.
Proof of Corollary 1.2. By Theorem 1.2 and Lemma 2.6, we have and , which imply Corollary 1.2.
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This work was supported by NSF of China under grant No. 11071069.
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Qian, WM. Schur convexity for the ratios of the Hamy and generalized Hamy symmetric functions. J Inequal Appl 2011, 131 (2011). https://doi.org/10.1186/1029-242X-2011-131
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DOI: https://doi.org/10.1186/1029-242X-2011-131