Abstract
The Schur-convexity for certain compound functions involving the dual of the complete symmetric function is studied. As an application, the Schur-convexity of some special symmetric functions is discussed and some inequalities are established.
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1 Introduction
Throughout the article, \(\mathbb{R}\) denotes the set of real numbers, \(\boldsymbol {x} = (x_{1}, x_{2}, \ldots , x_{n})\) denotes n-tuple (n-dimensional real vectors), the set of vectors can be written as
In particular, the notations \(\mathbb{R}\) and \(\mathbb{R}_{+}\) denote \(\mathbb{R}^{1}\) and \(\mathbb{R}^{1}_{+}\), respectively.
In recent years, the Schur-convexity, Schur-geometric, and Schur-harmonic convexities of various symmetric functions have been a hot topic of inequality research [1–30].
The following complete symmetric function is an important class of symmetric functions.
For \(\boldsymbol {x}=(x_{1},x_{2},\ldots ,x_{n}) \in \mathbb{R}^{n}\), the complete symmetric function \(c_{n}(\boldsymbol {x},r)\) is defined as
where \(c_{0}(\boldsymbol {x},r)=1\), \(r\in \{1,2,\ldots , n \}\), \(i_{1},i_{2},\ldots , i_{n}\) are nonnegative integers.
It has been investigated by many mathematicians, and there are many interesting results in the literature.
Guan [4] discussed the Schur-convexity of \(c_{n}(\boldsymbol {x},r)\) and proved the following.
Proposition 1
\(c_{n}(\boldsymbol {x},r)\)is increasing and Schur-convex on\(\mathbb{R}^{n}_{+}\).
Subsequently, Chu et al. [1] proved the following.
Proposition 2
\(c_{n}(\boldsymbol {x},r)\)is Schur-geometrically convex and Schur-harmonically convex on \(\mathbb{R}^{n}_{+}\).
In 2016, Shi et al. [18] further considered the Schur-convexity of \(c_{n}(\boldsymbol {x},r)\) on \(\mathbb{R}^{n}_{-}\), which proved the following proposition.
Proposition 3
Ifris an even integer (or odd integer, respectively), then\(c_{n}(\boldsymbol {x},r)\)is decreasing and Schur-convex (or increasing and Schur-concave, respectively) on\(\mathbb{R}^{n}_{-}\).
The dual form of the complete symmetric function \(c_{n}(\boldsymbol {x},r)\) is defined as
where \(c^{*}_{0}(\boldsymbol {x},r)=1\), \(r\in \{1,2,\ldots , n \}\), \(i_{1},i_{2},\ldots , i_{n}\) are nonnegative integers.
Zhang and Shi [17] proved the following two propositions.
Proposition 4
For\(r=1, 2,\ldots , n\), \(c^{*}_{n}(\boldsymbol {x},r)\)is increasing and Schur-concave on\(\mathbb{R}^{n}_{+}\).
Proposition 5
For\(r=1, 2,\ldots , n\), \(c^{*}_{n}(\boldsymbol {x},r)\)is Schur-geometrically convex and Schur-harmonically convex on\(\mathbb{R}^{n}_{+}\).
Notice that
it is not difficult to prove the following proposition.
Proposition 6
Ifris an even integer (or odd integer, respectively), then\(c^{*}_{n}(\boldsymbol {x},r)\)is decreasing and Schur-concave (or increasing and Schur-convex, respectively) on\(\mathbb{R}^{n}_{-}\).
In this paper we will study the Schur-convexity, Schur-geometric and Schur-harmonic convexities of the following composite function of \(c^{*}_{n} (\boldsymbol {x},r )\):
where f is a positive function which satisfies certain conditions.
Our main results are as follows.
Theorem 1
Let\(I \subset \mathbb{R}\)be a symmetric convex set with nonempty interior, and let\(f : I\rightarrow \mathbb{R}_{+}\)be continuous onIand differentiable in the interior ofI.
- (a)
Iffis a log-convex function onI, then for any\(r = 1,2, \ldots , n\), \(c^{*}_{n} (f(\boldsymbol {x}), r )\)is a Schur-convex function on\(I^{n}\);
- (b)
Iffis a concave function onI, then for any\(r = 1,2, \ldots , n\), \(c^{*}_{n} (f(\boldsymbol {x}), r )\)is a Schur-concave function on\(I^{n}\).
Theorem 2
Let\(I \subset \mathbb{R}_{+}\)be a symmetric convex set with nonempty interior and let\(f : I\rightarrow \mathbb{R}_{+}\)be continuous onIand differentiable in the interior ofI.
- (a)
Iffis an increasing and log-convex function onI, then for any\(r = 1,2, \ldots , n\), \(c^{*}_{n} (f(\boldsymbol {x}), r )\)is a Schur-geometrically convex function on\(I^{n}\).
- (b)
Iffis a descending and concave function onI, then for any\(r = 1,2, \ldots , n\), \(c^{*}_{n} (f(\boldsymbol {x}), r )\)is a Schur-geometrically concave function on\(I^{n}\).
Theorem 3
Let\(I \subset \mathbb{R}_{+}\)be a symmetric convex set with nonempty interior, and let\(f : I\rightarrow \mathbb{R}_{+}\)be continuous onIand differentiable in the interior ofI.
- (a)
Iffis an increasing and log-convex function onI, then for any\(r = 1,2, \ldots , n\), \(c^{*}_{n} (f(\boldsymbol {x}), r )\)is a Schur-harmonically convex function on\(I^{n}\).
- (b)
Iffis a descending and concave function onI, then for any\(r = 1,2, \ldots , n\), \(c^{*}_{n} (f(\boldsymbol {x}), r )\)is a Schur-harmonically concave function on\(I^{n}\).
2 Definitions and lemmas
For convenience, we introduce some definitions as follows.
Definition 1
Let \(\boldsymbol {x} = ( x_{1},x_{2},\ldots , x_{n })\) and \(\boldsymbol {y} = ( y_{1},y_{2},\ldots , y_{n }) \in \mathbb{R}^{n}\).
- (a)
\(\boldsymbol {x}\ge \boldsymbol {y}\) means \(x_{i} \ge y_{i}\) for all \(i=1, 2, \ldots , n\).
- (b)
Let \(\varOmega \subset \mathbb{R} ^{n}\), φ: \(\varOmega \to \mathbb{\mathbb{R}}\) is said to be increasing if \(\boldsymbol {x} \ge \boldsymbol {y}\) implies \(\varphi {(\boldsymbol {x})} \ge \varphi {(\boldsymbol {y})}\). φ is said to be decreasing if and only if −φ is increasing.
Definition 2
Let \(\boldsymbol {x} = ( x_{1},x_{2},\ldots , x_{n })\) and \(\boldsymbol {y} = ( y_{1},y_{2},\ldots , y_{n }) \in \mathbb{R}^{n}\).
- (a)
x is said to be majorized by y (in symbols \(\boldsymbol {x} \prec \boldsymbol {y}\)) if \(\sum_{i = 1}^{k} x_{[i]} \le \sum_{i = 1}^{k} y_{[i]}\) for \(k = 1,2,\ldots ,n - 1\) and \(\sum_{i = 1}^{n} x_{i} = \sum_{i = 1}^{n} y_{i}\), where \(x_{[1]}\ge x_{[2]}\ge \cdots \ge x_{[n]}\) and \(y_{[1]}\ge y_{[2]}\ge \cdots \ge y_{[n]}\) are rearrangements of x and y in a descending order.
- (b)
Let \(\varOmega \subset \mathbb{R}^{n}\), φ: \(\varOmega \to \mathbb{R}\) is said to be a Schur-convex function on Ω if \(\boldsymbol {x} \prec \boldsymbol {y}\) on Ω implies \(\varphi ( \boldsymbol {x} ) \le \)\(\varphi ( \boldsymbol {y} ) \). φ is said to be a Schur-concave function on Ω if and only if −φ is Schur-convex function on Ω.
Definition 3
Let \(\boldsymbol {x} = ( x_{1},x_{2},\ldots , x_{n })\) and \(\boldsymbol {y} = ( y_{1},y_{2},\ldots , y_{n }) \in \mathbb{R}^{n}\).
- (a)
\(\varOmega \subset \mathbb{R}^{n}\) is said to be a convex set if \(\boldsymbol {x},\boldsymbol {y}\in \varOmega \), \(0 \leq \alpha \leq 1\), implies \(\alpha \boldsymbol {x}+(1-\alpha )\boldsymbol {y}= (\alpha x_{1}+(1-\alpha )y_{1}, \alpha x_{2}+(1-\alpha )y_{2},\ldots ,\alpha x_{n}+(1-\alpha )y_{n} )\in \varOmega \).
- (b)
Let \(\varOmega \subset \mathbb{R}^{n}\) be a convex set. A function φ: \(\varOmega \to \mathbb{R}\) is said to be a convex function on Ω if
$$ \varphi \bigl(\alpha \boldsymbol {x}+(1-\alpha )\boldsymbol {y} \bigr)\leq \alpha \varphi ( \boldsymbol {x})+(1-\alpha )\varphi (\boldsymbol {y}) $$for all \(\boldsymbol {x},\boldsymbol {y}\in \varOmega \), and all \(\alpha \in [0,1]\). φ is said to be a concave function on Ω if and only if −φ is a convex function on Ω.
Definition 4
-
(a)
A set \(\varOmega \subset \mathbb{R}^{n}\) is called a symmetric set if \(\boldsymbol {x}\in \varOmega \) implies \(\boldsymbol {x}P \in \varOmega \) for every \(n\times n\) permutation matrix P.
-
(b)
A function \(\varphi : \varOmega \to \mathbb{R}\) is called symmetric if, for every permutation matrix P, \(\varphi (\boldsymbol {x}P) = \varphi (\boldsymbol {x})\) for all \(\boldsymbol {x} \in \varOmega \).
Lemma 1
(Schur-convex function decision theorem [31, 32])
Let\(\varOmega \subset \mathbb{R} ^{n} \)be symmetric and have a nonempty interior convex set. \(\varOmega ^{0}\)is the interior ofΩ. \(\varphi :\varOmega \to \mathbb{R} \)is continuous onΩand differentiable in\(\varOmega ^{0}\). Thenφis the Schur-convex (or Schur-concave, respectively) function if and only ifφis symmetric onΩand
holds for any\(\boldsymbol {x} \in \varOmega ^{0} \).
The first systematical study of the functions preserving the ordering of majorization was made by Issai Schur in 1923. In Schur’s honor, such functions are said to be “Schur-convex”. They can be used extensively in analytic inequalities, combinatorial optimization, quantum physics, information theory, and other related fields. See [31].
Definition 5
([33])
Let \(\boldsymbol {x} = ( x_{1},x_{2},\ldots , x_{n }) \in \mathbb{R}_{+}^{n}\) and \(\boldsymbol {y} = ( y_{1},y_{2},\ldots , y_{n }) \in \mathbb{R}_{+}^{n}\).
- (a)
\(\varOmega \subset \mathbb{R}_{+} ^{n}\) is called a geometrically convex set if \((x_{1}^{\alpha }y_{1}^{\beta },x_{2}^{\alpha }y_{2}^{\beta },\ldots ,x_{n}^{ \alpha }y_{n}^{\beta }) \in \varOmega \) for all x, \(\boldsymbol {y} \in \varOmega \) and α, \(\beta \in [0, 1]\) such that \(\alpha +\beta =1\).
- (b)
Let \(\varOmega \subset \mathbb{R}_{+} ^{n}\). The function φ: \(\varOmega \to \mathbb{R}_{+}\) is said to be a Schur-geometrically convex function on Ω if \((\log x_{1},\log x_{2},\ldots ,\log x_{n}) \prec (\log y_{1},\log y_{2}, \ldots , \log y_{n})\) on Ω implies \(\varphi (\boldsymbol {x} ) \le \varphi (\boldsymbol {y} )\). The function φ is said to be a Schur-geometrically concave function on Ω if and only if −φ is a Schur-geometrically convex function on Ω.
The Schur-geometric convexity was proposed by Zhang [33] in 2004, and it was investigated by Chu et al. [34], Guan [35], Sun et al. [36], and so on. We also note that some authors use the term “Schur multiplicative convexity”.
In 2009, Chu ([1, 2, 37]) introduced the notion of Schur-harmonically convex function, and some interesting inequalities were obtained.
Definition 6
([37])
Let \(\varOmega \subset \mathbb{R}_{+}^{n}\) or \(\varOmega \subset \mathbb{R}_{-}^{n}\).
- (a)
A set Ω is said to be harmonically convex if \(\frac{\boldsymbol{xy}}{\lambda {\boldsymbol{x}}+(1-\lambda ){\boldsymbol{y}}} \in \varOmega \) for every \({\boldsymbol{x},\boldsymbol{y}}\in \varOmega \) and \(\lambda \in [0,1]\), where \(\boldsymbol{xy}=\sum_{i=1}^{n}x_{i}y_{i}\) and \(\frac{1}{\boldsymbol{x}}= (\frac{1}{x_{1}}, \frac{1}{x_{2}}, \ldots ,\frac{1}{x_{n}} )\).
- (b)
A function \(\varphi :\varOmega \to \mathbb{R}_{+}\) is said to be Schur-harmonically convex on Ω if \(\frac{1}{\boldsymbol{x}} \prec \frac{1}{\boldsymbol{y}}\) implies \(\varphi ({\boldsymbol{x}}) \le \varphi ({\boldsymbol{y}})\). 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.
Remark 1
We extend the definition and determination theorem of Schur-harmonically convex function established by Chu as follows:
- (a)
\(\varOmega \subset \mathbb{R}^{n}_{+}\) is extended to \(\varOmega \subset \mathbb{R}^{n}_{+}\) or \(\varOmega \subset \mathbb{R}^{n}_{-}\);
- (b)
The function \(\varphi :\varOmega \to \mathbb{R}\) must not be a positive function.
Lemma 2
Let the set\(\mathbb{A}, \mathbb{B}\subset \mathbb{R}\), \(\varphi :\mathbb{B}^{n}\rightarrow \mathbb{R}\), \(f:\mathbb{A}\rightarrow \mathbb{B}\)and\(\psi (x_{1}, x_{2}, \ldots , x_{n}) = \varphi (f(x_{1}),f(x_{2}), \ldots , f(x_{n})):\mathbb{A}^{n}\rightarrow \mathbb{R}\).
- (a)
Iffis convex andφis increasing and Schur-convex, thenψis Schur-convex;
- (b)
Iffis concave, φis increasing and Schur-concave, thenψis Schur-concave.
Lemma 3
Let the set\(\varOmega \subset \mathbb{R}^{n}_{+}\). The function\(\varphi :\varOmega \rightarrow \mathbb{R}_{+}\)is differentiable.
- (a)
Ifφis increasing and Schur-convex, thenφis Schur geometrically convex.
- (b)
Ifφis decreasing and Schur-concave, thenφis Schur geometrically concave.
Lemma 4
Let the set\(\varOmega \subset \mathbb{R}^{n}_{+}\). The function\(\varphi :\varOmega \rightarrow \mathbb{R}_{+}\)is differentiable.
- (a)
Ifφis increasing and Schur-convex, thenφis Schur-harmonically convex.
- (b)
Ifφis decreasing and Schur-concave, thenφis Schur-harmonically concave.
Lemma 5
Let\((\boldsymbol {x} =(x_{1},x_{2}, \ldots ,x_{n})\in \mathbb{R}^{n}\). Then
where\(A(\boldsymbol {x})=\frac{1}{n}\sum^{n}_{i} x_{i}\).
Lemma 6
([22])
Let
If\(u>1\), then\(q(t)\)is a log-convex function on\(\mathbb{R}_{+}\).
3 Proof of main results
Proof of Theorem 1
For the case of \(r=1\) and \(r=2\), it is easy to prove that \(c^{*}_{n} (f(\boldsymbol {x}), r )\) is Schur-convex on \(I^{n}\).
Now consider the case of \(r \geq 3\). By the symmetry of \(c^{*}_{n} (f(\boldsymbol {x}), r )\), without loss of generality, we can set \(x_{1}> x_{2}\).
Then
By the same arguments,
where
and
where
- (a)
Since the log-convex function must be convex function, so \(f'(x_{1})-f'(x_{2})\geq 0\) and \(f(x_{2})f'(x_{1})-f(x_{1})f'(x_{2})\geq 0\), and since \((f(x)f'(x))'= (f'(x))^{2}+f(x)f''(x)\geq 0\), so \(f(x_{1})f'(x_{1})-f(x_{2})f'(x_{2})\geq 0\), and then \(A_{1} \geq 0\) and \(A_{2} \geq 0\). For \(\boldsymbol {x}\in I^{n}\), we have
$$ \frac{\partial c^{*}_{n} (f(\boldsymbol {x}), r )}{\partial x_{1}}- \frac{\partial c^{*}_{n} (f(\boldsymbol {x}), r )}{\partial x_{2}} \geq 0, $$by Lemma 1, it follows that \(c^{*}_{n} (f(\boldsymbol {x}), r )\) is Schur-convex on \(I^{n}\).
- (b)
By Proposition 4, we know that \(c^{*}_{n}(\boldsymbol {x},r)\) is increasing and Schur-concave on \(\mathbb{R}^{n}_{+}\). Since f is concave, from (b) in Lemma 4 it follows that \(c^{*}_{n} (f(\boldsymbol {x}), r )\) is Schur-concave on \(I^{n}\).
The proof of Theorem 1 is completed. □
Proof of Theorem 2
Theorem 2 can be proved by Theorem 1 combined with Lemma 3.
The proof of Theorem 2 is completed. □
Proof of Theorem 3
Theorem 3 can be proved by Theorem 1 combined with Lemma 4.
The proof of Theorem 3 is completed. □
4 Applications
Let
Theorem 4
The symmetric function\(c^{*}_{n} (\frac{1}{\boldsymbol {x}},r )\)is Schur-convex on\(\mathbb{R}^{n}_{+}\). Ifris an even integer (or odd integer, respectively ), then\(c^{*}_{n} (\frac{1}{\boldsymbol {x}},r )\)is Schur-convex (or Schur-concave, respectively) on\(\mathbb{R}^{n}_{-}\).
Proof
Let \(f(x)=\frac{1}{x}\). Then \((\ln f(x))'' = \frac{1}{x^{2}}\), so \(f(x)\) is log-convex on \(\mathbb{R}_{+}\), by (a) in Theorem 1, it follows that \(c^{*}_{n} (\frac{1}{\boldsymbol {x}},r )\) is Schur-convex on \(\mathbb{R}^{n}_{+}\).
For \(\boldsymbol {x}\in \mathbb{R}^{n}_{-}\), \(-\boldsymbol {x}\in \mathbb{R}^{n}_{+}\), so \(c^{*}_{n} (\frac{1}{-\boldsymbol {x}},r )\) is Schur-convex on \(\mathbb{R}^{n}_{-}\). But
This means that if r is an even integer, then
is Schur-convex on \(\mathbb{R}^{n}_{-}\).
If r is an odd integer, then
is Schur-concave on \(\mathbb{R}^{n}_{-}\).
The proof of Theorem 4 is completed. □
By Theorem 4 and majorizing relation (7), it is not difficult to prove the following corollary.
Corollary 1
If\(\boldsymbol {x} \in \mathbb{R}^{n}_{+}\)orris an even integer and\(\boldsymbol {x} \in \mathbb{R}^{n}_{-}\), then we have
where\(A_{n}(\boldsymbol {x})= \frac{1}{n}\sum^{n}_{i=1}x_{i}\)and. Ifris odd and\(\boldsymbol {x} \in \mathbb{R}^{n}_{-}\), then inequality (10) is reversed.
Let
Theorem 5
The symmetric function\(c^{*}_{n} (\frac{\boldsymbol {x}}{1-\boldsymbol {x}},r )\)is Schur-convex, Schur-geometrically convex, and Schur-harmonically convex on\([\frac{1}{2}, 1]^{n}\).
Proof
Let \(g(x)=\frac{x}{1-x}\). Then \((\ln g(x))'' = \frac{2x-1}{x^{2}(1-x)^{2}}\), so \(f(x)\) is log-convex on \([\frac{1}{2}, 1]\); by Theorem 1(a), it follows that \(c^{*}_{n} (\frac{\boldsymbol {x}}{1-\boldsymbol {x}},r )\) is Schur-convex on \([\frac{1}{2}, 1]^{n}\). Noting that \(g(x)\) is increasing on \([\frac{1}{2}, 1]\), by (a) in Theorem 2 and (a) in Theorem 3, it follows that \(c^{*}_{n} (\frac{\boldsymbol {x}}{1-\boldsymbol {x}},r )\) is Schur-geometrically convex and Schur-harmonically convex on \([\frac{1}{2}, 1]^{n}\).
The proof of Theorem 5 is completed. □
From the majorizing relation (7), the following majorizing relation is established:
By this majorizing relation and Theorem 5, it is not difficult to prove the following corollary.
Corollary 2
If\(\boldsymbol {x} \in [\frac{1}{2}, 1]^{n}\), then we have
where\(G_{n}(\boldsymbol {x})=\sqrt[n]{\prod^{n}_{i=1}x_{i}}\).
Let
Theorem 6
-
(a)
The symmetric function\(c^{*}_{n} (\frac{1+\boldsymbol {x}}{1-\boldsymbol {x}},r )\)is Schur-convex, Schur-geometrically convex, and Schur-harmonically convex on\((0,1)^{n}\).
-
(b)
Ifris an even integer (or odd integer, respectively ), then\(c^{*}_{n} (\frac{1+\boldsymbol {x}}{1-\boldsymbol {x}},r )\)is Schur-convex (or Schur-concave, respectively) on\((1, +\infty )^{n}\).
Proof
(a) Let \(h(x)=\frac{1+x}{1-x}\). Then \((\ln h(x))'' = \frac{4x}{(1+x)^{2}(1-x)^{2}}\), so \(f(x)\) is log-convex on \((0,1)\), by Theorem 1(a), it follows that \(c^{*}_{n} (\frac{1+\boldsymbol {x}}{1-\boldsymbol {x}},r )\) is Schur-convex on \((0,1)^{n}\). Noting that \(h(x)\) is increasing on \((0,1)^{n}\), by (a) in Theorem 2 and (a) in Theorem 3, it follows that \(c^{*}_{n} (\frac{1+\boldsymbol {x}}{1-\boldsymbol {x}},r )\) is Schur-geometrically convex and Schur-harmonically convex on \((0,1)^{n}\).
(b) For \(\boldsymbol {x} \in (1, + \infty )\), we consider
Let \(h_{1}(x)=\frac{1+x}{x-1}\). Then \((\ln h_{1}(x))'' = \frac{4x}{(1+x)^{2}( x-1)^{2}}\), so \(f(x)\) is log-convex on \((1, + \infty )\), by (a) in Theorem 1, it follows that \(c^{*}_{n} (\frac{1+\boldsymbol {x}}{\boldsymbol {x}-1},r )\) is Schur-convex on \((1, + \infty )^{n}\).
Noting that
combining the Schur-convexity of \(c^{*}_{n} (\frac{1+\boldsymbol {x}}{\boldsymbol {x}-1},r )\), we can get (b) in Theorem 6.
The proof of Theorem 6 is completed. □
Let
Theorem 7
-
(a)
Ifris an even integer (or odd integer, respectively), then\(c^{*}_{n} (\frac{1}{\boldsymbol {x}}-\boldsymbol {x},r )\)is Schur-concave (or Schur-convex, respectively) on\(\mathbb{R}^{n}_{+}\).
-
(b)
The symmetric function\(c^{*}_{n} (\frac{1}{\boldsymbol {x}}-\boldsymbol {x},r )\)is Schur-concave on\(\mathbb{R}^{n}_{-}\).
-
(c)
Ifris an even integer, then\(c^{*}_{n} (\frac{1}{\boldsymbol {x}}-\boldsymbol {x},r )\)is Schur-geometrically concave and Schur-harmonically concave on\((-\infty ,1]^{n}\).
Proof
First consider
- (a)
Let \(p(x)=x-\frac{1}{x}\). Then \(p''(x) = -\frac{2}{x^{3}}\), so \(f(x)\) is concave on \(\mathbb{R}_{+}\), by Theorem 1(b), it follows that \(c^{*}_{n} (\boldsymbol {x}-\frac{1}{\boldsymbol {x}},r )\) is Schur-concave on \(\mathbb{R}^{n}_{+}\).
Noting that
$$ c^{*}_{n} \biggl(\frac{1}{\boldsymbol {x}}-\boldsymbol {x},r \biggr)=(-1)^{n} c^{*}_{n} \biggl(\boldsymbol {x}- \frac{1}{\boldsymbol {x}},r \biggr), $$combining the Schur-concavity of \(c^{*}_{n} (\frac{1}{\boldsymbol {x}}-\boldsymbol {x},r )\), we can get (a) in Theorem 7.
- (b)
Noting that
$$ c^{*}_{n} \biggl(\frac{1}{-\boldsymbol {x}}-(-\boldsymbol {x}),r \biggr)= (-1)^{r} c^{*}_{n} \biggl( \frac{1}{\boldsymbol {x}}-\boldsymbol {x},r \biggr), $$combining (a) in Theorem 7, it is not difficult to verify that (b) in Theorem 7 holds.
- (c)
It is not difficult to verify that \(p(x)=x-\frac{1}{x}\) is nonnegative and decreasing on \((-\infty , 1]\), by Lemma 5 and Lemma 6, from (a) and (b) in Theorem 7, it follows that (c) in Theorem 7 holds.
The proof of Theorem 7 is completed. □
For \(u >1\), let
Theorem 8
The symmetric function\(c^{*}_{n} (\frac{u^{\boldsymbol {x}}-1}{\boldsymbol {x}},r )\)is Schur-convex, Schur-geometrically convex, and Schur-harmonically convex on\(\mathbb{R}^{n}_{+}\)for\(u>1\).
Proof
Let \(q(t)=\frac{u^{t}-1}{t}\). Then from Lemma 6 and (a) in Theorem 1, it follows that \(c^{*}_{n} (\frac{u^{\boldsymbol {x}}-1}{\boldsymbol {x}},r )\) is Schur-convex on \(\mathbb{R}^{n}_{+}\) for \(u>1\).
Since
where \(s(t)=u^{t}(t\log u-1)+1\), \(s'(t)=u^{t}\log u\log u^{t}>0\), for \(u>1\) and \(t>0\), so \(s(t)\geq s(0)=0\), and then \(q'(t)\geq 0\), that is, \(q(t)\) is increasing on \(\mathbb{R}^{n}_{+}\), by (a) in Theorem 2 and (a) in Theorem 3, it follows that \(c^{*}_{n} (\frac{u^{\boldsymbol {x}}-1}{\boldsymbol {x}},r )\) is Schur-geometrically convex and Schur-harmonically convex on \(\mathbb{R}^{n}_{+}\).
The proof of Theorem 8 is completed. □
From the majorizing relation (7), the following majorizing relation is established:
By this majorizing relation and Theorem 8, it is not difficult to prove the following corollary.
Corollary 3
If\(\boldsymbol {x}=(x_{1}, x_{2},\ldots ,x_{n}) \in \mathbb{R}^{n}_{+}\)and\(u>1\), then
where\(H_{n}(\boldsymbol {x})= \frac{n}{\sum^{n}_{i=1}x^{-1}_{i}}\).
Discovering and judging Schur convexity of various symmetric functions is an important subject in the study of the majorization theory. In recent years, many domestic scholars have made a lot of achievements in this field (see [24–30]).
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Shi, HN., Wang, P. & Zhang, J. Schur-convexity for compositions of complete symmetric function dual. J Inequal Appl 2020, 65 (2020). https://doi.org/10.1186/s13660-020-02334-8
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DOI: https://doi.org/10.1186/s13660-020-02334-8