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.

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

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

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 [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

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

(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.

• (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

([31, 32])

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

• (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

([31, 32])

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

• (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

([31, 32])

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

• (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

([31, 32])

• (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

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

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

• (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

([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$\psi (x_{1}, x_{2}, \ldots , x_{n}) = \varphi (f(x_{1}),f(x_{2}), \ldots , f(x_{n})):\mathbb{A}^{n}\rightarrow \mathbb{R}$.

• (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

([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$\mathbb{R}_{+}$.

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

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

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

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

• (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

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

(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$\left(\begin{array}{c}n+r-1\\ r\end{array}\right)=\frac{(n+r-1)!}{r!((n+r-1)-r)!}$. 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

• (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

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

(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

• (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

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

• (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

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

(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 [24–30]).