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# Schur-convexity for compositions of complete symmetric function dual

## 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.

## 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

\begin{aligned}& \mathbb{R}^{n} = \bigl\{ {\boldsymbol {x}=(x_{1}, x_{2}, \ldots , x_{n}): x_{i} \in \mathbb{R}, i = 1,2,\ldots ,n} \bigr\} , \\& \mathbb{R}^{n}_{+}=\bigl\{ \boldsymbol {x}=(x_{1},x_{2}, \ldots ,x_{n}): x_{i}>0, i=1,2, \ldots ,n\bigr\} , \\& \mathbb{R}^{n}_{-}=\bigl\{ \boldsymbol {x}=(x_{1},x_{2}, \ldots ,x_{n}): x_{i}< 0, i=1,2, \ldots ,n\bigr\} . \end{aligned}

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 [130].

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

$$c_{n}(\boldsymbol {x},r)=\sum_{i_{1}+ i_{2}+\cdots +i_{n}=r}x_{1}^{i_{1}}x_{2}^{i_{2}} \cdots x_{n}^{i_{n}},$$
(1)

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

$$c^{*}_{n}(\boldsymbol {x},r)=\prod_{i_{1}+ i_{2}+\cdots +i_{n}=r} \sum^{n}_{j=1}i_{j} x_{j},$$
(2)

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

$$c^{*}_{n}(\boldsymbol {-x},r)=(-1)^{r} c^{*}_{n}(\boldsymbol {x},r),$$

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 $$c^{*}_{n} (\boldsymbol {x},r )$$:

$$c^{*}_{n} \bigl(f(\boldsymbol {x}),r \bigr)=c^{*}_{n} \bigl(f(x_{1}),f(x_{2}), \ldots, f(x_{n}),r \bigr)=\prod_{i_{1}+ i_{2}+\cdots +i_{n}=r} \sum ^{n}_{ j=1}i_{j} \bigl(f(x_{j}) \bigr),$$
(3)

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.

1. (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$$I^{n}$$;

2. (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.

1. (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$$I^{n}$$.

2. (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.

1. (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$$I^{n}$$.

2. (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$$I^{n}$$.

## Definitions and lemmas

For convenience, we introduce some definitions as follows.

### Definition 1

([31, 32])

Let $$\boldsymbol {x} = ( x_{1},x_{2},\ldots , x_{n })$$ and $$\boldsymbol {y} = ( y_{1},y_{2},\ldots , y_{n }) \in \mathbb{R}^{n}$$.

1. (a)

$$\boldsymbol {x}\ge \boldsymbol {y}$$ means $$x_{i} \ge y_{i}$$ for all $$i=1, 2, \ldots , n$$.

2. (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

([31, 32])

Let $$\boldsymbol {x} = ( x_{1},x_{2},\ldots , x_{n })$$ and $$\boldsymbol {y} = ( y_{1},y_{2},\ldots , y_{n }) \in \mathbb{R}^{n}$$.

1. (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.

2. (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

([31, 32])

Let $$\boldsymbol {x} = ( x_{1},x_{2},\ldots , x_{n })$$ and $$\boldsymbol {y} = ( y_{1},y_{2},\ldots , y_{n }) \in \mathbb{R}^{n}$$.

1. (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 $$\alpha \boldsymbol {x}+(1-\alpha )\boldsymbol {y}= (\alpha x_{1}+(1-\alpha )y_{1}, \alpha x_{2}+(1-\alpha )y_{2},\ldots ,\alpha x_{n}+(1-\alpha )y_{n} )\in \varOmega$$.

2. (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

([31, 32])

1. (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.

2. (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

$$( x_{1} - x_{2} ) \biggl( \frac{\partial \varphi }{\partial x_{1}} - \frac{\partial \varphi }{\partial x_{2} } \biggr) \ge 0\ (\textit{or }\leq 0, \textit{respectively})$$
(4)

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 $$\boldsymbol {y} = ( y_{1},y_{2},\ldots , y_{n }) \in \mathbb{R}_{+}^{n}$$.

1. (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 $$\alpha +\beta =1$$.

2. (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 $$\varOmega \subset \mathbb{R}_{-}^{n}$$.

1. (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 $$\frac{1}{\boldsymbol{x}}= (\frac{1}{x_{1}}, \frac{1}{x_{2}}, \ldots ,\frac{1}{x_{n}} )$$.

2. (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:

1. (a)

$$\varOmega \subset \mathbb{R}^{n}_{+}$$ is extended to $$\varOmega \subset \mathbb{R}^{n}_{+}$$ or $$\varOmega \subset \mathbb{R}^{n}_{-}$$;

2. (b)

The function $$\varphi :\varOmega \to \mathbb{R}$$ must not be a positive function.

### Lemma 2

([31, 32])

Let the set$$\mathbb{A}, \mathbb{B}\subset \mathbb{R}$$, $$\varphi :\mathbb{B}^{n}\rightarrow \mathbb{R}$$, $$f:\mathbb{A}\rightarrow \mathbb{B}$$and$$\psi (x_{1}, x_{2}, \ldots , x_{n}) = \varphi (f(x_{1}),f(x_{2}), \ldots , f(x_{n})):\mathbb{A}^{n}\rightarrow \mathbb{R}$$.

1. (a)

Iffis convex andφis increasing and Schur-convex, thenψis Schur-convex;

2. (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.

1. (a)

Ifφis increasing and Schur-convex, thenφis Schur geometrically convex.

2. (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.

1. (a)

Ifφis increasing and Schur-convex, thenφis Schur-harmonically convex.

2. (b)

Ifφis decreasing and Schur-concave, thenφis Schur-harmonically concave.

### Lemma 5

([31, 32])

Let$$(\boldsymbol {x} =(x_{1},x_{2}, \ldots ,x_{n})\in \mathbb{R}^{n}$$. Then

$$\bigl(A(\boldsymbol {x}),A(\boldsymbol {x}), \ldots , A(\boldsymbol {x})\bigr)\prec (\boldsymbol {x} =(x_{1},x_{2}, \ldots ,x_{n}),$$
(5)

where$$A(\boldsymbol {x})=\frac{1}{n}\sum^{n}_{i} x_{i}$$.

### Lemma 6

([22])

Let

$$q(t)=\frac{u^{t}-1}{t}.$$

If$$u>1$$, then$$q(t)$$is a log-convex function on$$\mathbb{R}_{+}$$.

## 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 $$x_{1}> x_{2}$$.

\begin{aligned} c^{*}_{n} \bigl((\boldsymbol {x}), r \bigr) ={}& \prod _{{i_{1}+i_{2}+\cdots +i_{n}=r \atop i_{1}\neq 0, i_{2}=0 }} {\sum_{j=1}^{n}i_{j}f(x_{j})} \times \prod_{{i_{1}+i_{2}+\cdots +i_{n}=r \atop i_{1}= 0, i_{2}\neq 0 }} {\sum _{j=1}^{n}i_{j}f(x_{j})} \\ &{} \times \prod_{{i_{1}+i_{2}+\cdots +i_{n}=r \atop i_{1}\neq 0, i_{2} \neq 0 }} {\sum _{j=1}^{n}i_{j}f(x_{j})} \times \prod_{{i_{1}+i_{2}+ \cdots +i_{n}=r \atop i_{1}= 0, i_{2}=0 }} {\sum _{j=1}^{n}i_{j}f(x_{j})}. \end{aligned}

Then

\begin{aligned} \frac{\partial c^{*}_{n} (f(\boldsymbol {x}), r )}{\partial x_{1}}= {}&c^{*}_{n} \bigl(f(\boldsymbol {x}), r \bigr) \\ &{}\times \biggl( \sum_{{i_{1}+i_{2}+\cdots +i_{n}=r \atop i_{1}\neq 0, i_{2}=0 }} \frac{i_{1}f'(x_{1})}{\sum_{j=1}^{n}i_{j}f(x_{j})}+ \sum_{{i_{1}+i_{2}+ \cdots +i_{n}=r \atop i_{1}\neq 0, i_{2}\neq 0 }} \frac{i_{1}f'(x_{1})}{\sum_{j=1}^{n}i_{j}f(x_{j})} \biggr) \\ ={}& c^{*}_{n} \bigl(f(\boldsymbol {x}), r \bigr) \biggl( \sum _{{k+k_{3}+\cdots +k_{n}=r \atop k\neq 0}} \frac{kf'(x_{1})}{k f(x_{1})+\sum_{j=3}^{n}i_{j}f(x_{j})} \\ &{}+\sum_{{k+m+i_{3}+\cdots +i_{n}=r \atop k\neq 0, m\neq 0 }} \frac{kf'(x_{1})}{k f(x_{1})+m f(x_{2})+\sum_{j=3}^{n}i_{j}f(x_{j})} \biggr). \end{aligned}
(6)

By the same arguments,

\begin{aligned} \begin{aligned} &\frac{\partial c^{*}_{n} (f(\boldsymbol {x}), r )}{\partial x_{2}}= c^{*}_{n} \bigl(f(\boldsymbol {x}), r \bigr) \\ &\hphantom{\frac{\partial c^{*}_{n} (f(\boldsymbol {x}), r )}{\partial x_{2}}}= c^{*}_{n} \bigl(f(\boldsymbol {x}), r \bigr) \biggl( \sum _{{k+k_{3}+\cdots +k_{n}=r \atop k\neq 0}} \frac{kf'(x_{2})}{k f(x_{2})+\sum_{j=3}^{n}i_{j}f(x_{j})} \\ &\hphantom{\frac{\partial c^{*}_{n} (f(\boldsymbol {x}), r )}{\partial x_{2}}=}{}+\sum_{{k+m+i_{3}+\cdots +i_{n}=r \atop k\neq 0, m\neq 0 }} \frac{kf'(x_{2})}{k f(x_{2})+m f(x_{1})+\sum_{j=3}^{n}i_{j}f(x_{j})} \biggr), \\ &\frac{\partial c^{*}_{n} (f(\boldsymbol {x}), r )}{\partial x_{1}}- \frac{\partial c^{*}_{n} (f(\boldsymbol {x}), r )}{\partial x_{2}}=c^{*}_{n} \bigl(f(\boldsymbol {x}), r \bigr) (A_{1}+A_{2}), \end{aligned} \end{aligned}
(7)

where

\begin{aligned} A_{1} &=\sum_{{k+k_{3}+\cdots +k_{n}=r \atop k\neq 0}} \biggl( \frac{kf'(x_{1})}{k f(x_{1})+\sum_{j=3}^{n}i_{j}f(x_{j})}- \frac{kf'(x_{2})}{k f(x_{2})+\sum_{j=3}^{n}i_{j}f(x_{j})} \biggr) \\ &= k\sum_{{k+k_{3}+\cdots +k_{n}=r \atop k\neq 0}} \frac{k(f(x_{2})f'(x_{1})-f(x_{1})f'(x_{2}))+(f'(x_{1})-f'(x_{2}))\sum_{j=3}^{n}i_{j}f(x_{j})}{(k f(x_{1})+\sum_{j=3}^{n}i_{j}f(x_{j}))(k f(x_{2})+\sum_{j=3}^{n}i_{j}f(x_{j}))} \end{aligned}
(8)

and

\begin{aligned} A_{2}&=\sum_{{k+m+i_{3}+\cdots +i_{n}=r \atop k\neq 0, m\neq 0 }} \biggl( \frac{kf'(x_{1})}{k f(x_{1})+m f(x_{2})+\sum_{j=3}^{n}i_{j}f(x_{j})}- \frac{kf'(x_{2})}{k f(x_{2})+m f(x_{1})+\sum_{j=3}^{n}i_{j}f(x_{j})} \biggr) \\ &= k \sum_{{k+m+i_{3}+\cdots +i_{n}=r \atop k\neq 0, m\neq 0 }} \frac{\delta }{(k f(x_{1})+m f(x_{2})+\sum_{j=3}^{n}i_{j}f(x_{j}))(k f(x_{2})+m f(x_{1})+\sum_{j=3}^{n}i_{j}f(x_{j}))} \end{aligned}

where

\begin{aligned} \delta ={}& k\bigl(f(x_{2})f'(x_{1})-f(x_{1})f'(x_{2}) \bigr)+ m\bigl(f(x_{1})f'(x_{1})-f(x_{2})f'(x_{2}) \bigr) \\ &{}+\bigl(f'(x_{1})-f'(x_{2}) \bigr) \sum_{j=3}^{n}i_{j}f(x_{j}). \end{aligned}
1. (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 $$I^{n}$$.

2. (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. □

## Applications

Let

$$c^{*}_{n} \biggl(\frac{1}{\boldsymbol {x}},r \biggr)=\prod _{i_{1}+ i_{2}+ \cdots +i_{n}=r}\sum^{n}_{ j=1}i_{j} \biggl(\frac{1}{x_{j}} \biggr).$$
(9)

### 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

$$c^{*}_{n} \biggl(\frac{1}{-\boldsymbol {x}},r \biggr)= (-1)^{r} c^{*}_{n} \biggl( \frac{1}{\boldsymbol {x}},r \biggr).$$

This means that if r is an even integer, then

$$c^{*}_{n} \biggl(\frac{1}{\boldsymbol {x}},r \biggr)= c^{*}_{n} \biggl( \frac{1}{-\boldsymbol {x}},r \biggr)$$

is Schur-convex on $$\mathbb{R}^{n}_{-}$$.

If r is an odd integer, then

$$c^{*}_{n} \biggl(\frac{1}{\boldsymbol {x}},r \biggr)= -c^{*}_{n} \biggl( \frac{1}{-\boldsymbol {x}},r \biggr)$$

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

$$\prod_{i_{1}+ i_{2}+\cdots +i_{n}=r}\sum^{n}_{ j=1}i_{j} \biggl(\frac{1}{x_{j}} \biggr)\geq \biggl(\frac{r}{A_{n}(\boldsymbol {x})} \biggr)^{\binom{n+r-1}{r}},$$
(10)

where$$A_{n}(\boldsymbol {x})= \frac{1}{n}\sum^{n}_{i=1}x_{i}$$and$\left(\begin{array}{c}n+r-1\\ r\end{array}\right)=\frac{\left(n+r-1\right)!}{r!\left(\left(n+r-1\right)-r\right)!}$. Ifris odd and$$\boldsymbol {x} \in \mathbb{R}^{n}_{-}$$, then inequality (10) is reversed.

Let

$$c^{*}_{n} \biggl(\frac{\boldsymbol {x}}{1-\boldsymbol {x}},r \biggr)=\prod _{i_{1}+ i_{2}+\cdots +i_{n}=r}\sum^{n}_{ j=1}i_{j} \biggl( \frac{x_{j}}{1-x_{j}} \biggr).$$
(11)

### 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:

$$\bigl(\log G_{n}(\boldsymbol {x}), \log G_{n}(\boldsymbol {x}), \ldots , \log G_{n}( \boldsymbol {x}) \bigr)\prec (\log x_{1}, \log x_{2},\ldots ,\log x_{n} ).$$

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

$$\prod_{i_{1}+ i_{2}+\cdots +i_{n}=r}\sum^{n}_{ j=1}i_{j} \biggl(\frac{x_{j}}{1-x_{j}} \biggr)\geq \biggl( \frac{rG_{n}(\boldsymbol {x})}{1-G_{n}(\boldsymbol {x})} \biggr)^{\binom{n+r-1}{r}},$$
(12)

where$$G_{n}(\boldsymbol {x})=\sqrt[n]{\prod^{n}_{i=1}x_{i}}$$.

Let

$$c^{*}_{n} \biggl(\frac{1+\boldsymbol {x}}{1-\boldsymbol {x}},r \biggr)=\prod _{i_{1}+ i_{2}+\cdots +i_{n}=r}\sum^{n}_{ j=1}i_{j} \biggl( \frac{1+x_{j}}{1-x_{j}} \biggr).$$
(13)

### Theorem 6

1. (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$$(0,1)^{n}$$.

2. (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

$$c^{*}_{n} \biggl(\frac{1+\boldsymbol {x}}{\boldsymbol {x}-1},r \biggr)=\prod _{i_{1}+ i_{2}+\cdots +i_{n}=r}\sum^{n}_{ j=1}i_{j} \biggl( \frac{1+x_{j}}{x_{j}-1} \biggr).$$
(14)

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

$$c^{*}_{n} \biggl(\frac{1+\boldsymbol {x}}{1-\boldsymbol {x}},r \biggr)=(-1)^{r} c^{*}_{n} \biggl( \frac{1+\boldsymbol {x}}{\boldsymbol {x}-1},r \biggr),$$

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

$$c^{*}_{n} \biggl(\frac{1}{\boldsymbol {x}}-\boldsymbol {x},r \biggr)= \prod_{i_{1}+ i_{2}+\cdots +i_{n}=r}\sum^{n}_{ j=1}i_{j} \biggl(\frac{1}{x_{j}}-x_{j} \biggr).$$
(15)

### Theorem 7

1. (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$$\mathbb{R}^{n}_{+}$$.

2. (b)

The symmetric function$$c^{*}_{n} (\frac{1}{\boldsymbol {x}}-\boldsymbol {x},r )$$is Schur-concave on$$\mathbb{R}^{n}_{-}$$.

3. (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

$$c^{*}_{n} \biggl(\boldsymbol {x}-\frac{1}{\boldsymbol {x}},r \biggr)= \prod_{i_{1}+ i_{2}+\cdots +i_{n}=r}\sum^{n}_{ j=1}i_{j} \biggl(x_{j}- \frac{1}{x_{j}} \biggr).$$
1. (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.

2. (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.

3. (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

$$c^{*}_{n} \biggl(\frac{u^{\boldsymbol {x}}-1}{\boldsymbol {x}},r \biggr)=\prod _{i_{1}+ i_{2}+\cdots +i_{n}=r}\sum^{n}_{ j=1}i_{j} \biggl( \frac{u^{x_{j}}-1}{x_{j}} \biggr).$$
(16)

### 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

$$q'(t)= \frac{s(t)}{t^{2}},$$

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:

$$\biggl(\frac{1}{H_{n}(\boldsymbol {x})}, \frac{1}{H_{n}(\boldsymbol {x})}, \ldots , \frac{1}{H_{n}(\boldsymbol {x})} \biggr)\prec \biggl(\frac{1}{x_{1}}, \frac{1}{x_{2}},\ldots ,\frac{1}{x_{n}} \biggr).$$

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

$$\prod_{i_{1}+ i_{2}+\cdots +i_{n}=r}\sum^{n}_{ j=1}i_{j} \biggl(\frac{u^{x_{j}}-1}{x_{j}} \biggr)\geq \biggl( \frac{r(u^{H_{n}(\boldsymbol {x})}-1)}{H_{n}(\boldsymbol {x})} \biggr)^{\binom{n+r-1}{r}},$$
(17)

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 [2430]).

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The paper was supported by the General Project of Science and Technology Plan of Beijing Municipal Education Commission under Grant No. KM202011417012.

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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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### Keywords

• Schur-convexity
• Schur-geometric convexity
• Schur-harmonic convexity
• Completely symmetric function
• Dual form