Some new judgement theorems of Schur geometric and Schur harmonic convexities for a class of symmetric functions

Abstract

The judgement theorems of Schur geometric and Schur harmonic convexities for a class of symmetric functions are given. As their application, some analytic inequalities are established.

MSC:26D15, 05E05, 26B25.

1 Introduction

Throughout this paper, ℝ denotes the set of real numbers, $\mathbf{x}=(x_1,x_2,\dots ,x_n)$ denotes n-tuple (n-dimensional real vectors), the set of vectors can be written as

$$\begin{array}{c}{\mathbb{R}}^n=\{\mathbf{x}=(x_1,\dots ,x_n):x_i\in \mathbb{R},i=1,\dots ,n\},\hfill \\ {\mathbb{R}}_{+}^n=\{\mathbf{x}=(x_1,\dots ,x_n):x_i>0,i=1,\dots ,n\}.\hfill \end{array}$$

In particular, the notations ℝ and ${\mathbb{R}}_{+}$ denote ${\mathbb{R}}^1$ and ${\mathbb{R}}_{+}^1$, respectively.

Let $\pi =(\pi (1),\dots ,\pi (n))$ be a permutation of $(1,\dots ,n)$, all permutations are totally n!. The following conclusion is proved in [[1], pp.127-129].

Theorem A Let $A\subset {\mathbb{R}}^k$ be a symmetric convex set, and let φ be a Schur-convex function defined on A with the property that for each fixed $x_2,\dots ,x_k$, $\phi (z,x_2,\dots ,x_k)$ is convex in z on $\{z:(z,x_2,\dots ,x_k)\in A\}$. Then, for any $n>k$,

$$\psi (x_1,\dots ,x_n)=\sum _{\pi }\phi (x_{\pi (1)},\dots ,x_{\pi (k)})$$

(1)

is Schur-convex on

$$B=\{(x_1,\dots ,x_n):(x_{\pi (1)},\dots ,x_{\pi (k)})\in A\mathit{\text{ for all permutations }}\pi \}.$$

Furthermore, the symmetric function

$$\overline{\psi }(\mathbf{x})=\sum _{1\le i_1<\cdots <i_k\le n}\phi (x_{i_1},\dots ,x_{i_k})$$

(2)

is also Schur-convex on B.

Theorem A is very effective for judgement of the Schur-convexity of the symmetric functions of the form (2), see the references [1] and [2].

The Schur geometrically convex functions were proposed by Zhang [3] in 2004. Further, the Schur harmonically convex functions were proposed by Chu and Lü [4] in 2009. The theory of majorization was enriched and expanded by using these concepts [5–15]. Regarding Schur geometrically convex functions and Schur harmonically convex functions, the aim of this paper is to establish the following judgement theorems which are similar to Theorem A.

Theorem 1 Let $A\subset {\mathbb{R}}^k$ be a symmetric geometrically convex set, and let φ be a Schur geometrically convex (concave) function defined on A with the property that for each fixed $x_2,\dots ,x_k$, $\phi (z,x_2,\dots ,x_k)$ is GA convex (concave) in z on $\{z:(z,x_2,\dots ,x_k)\in A\}$. Then, for any $n>k$,

$$\psi (x_1,\dots ,x_n)=\sum _{\pi }\phi (x_{\pi (1)},\dots ,x_{\pi (k)})$$

is Schur geometrically convex (concave) on

$$B=\{(x_1,\dots ,x_n):(x_{\pi (1)},\dots ,x_{\pi (k)})\in A\mathit{\text{ for all permutations }}\pi \}.$$

Furthermore, the symmetric function

$$\overline{\psi }(\mathbf{x})=\sum _{1\le i_1<\cdots <i_k\le n}\phi (x_{i_1},\dots ,x_{i_k})$$

is also Schur geometrically convex (concave) on B.

Theorem 2 Let $A\subset {\mathbb{R}}^k$ be a symmetric harmonically convex set, and let φ be a Schur harmonically convex (concave) function defined on A with the property that for each fixed $x_2,\dots ,x_k$, $\phi (z,x_2,\dots ,x_k)$ is HA convex (concave) in z on $\{z:(z,x_2,\dots ,x_k)\in A\}$. Then, for any $n>k$,

$$\psi (x_1,\dots ,x_n)=\sum _{\pi }\phi (x_{\pi (1)},\dots ,x_{\pi (k)})$$

is Schur harmonically convex (concave) on

$$B=\{(x_1,\dots ,x_n):(x_{\pi (1)},\dots ,x_{\pi (k)})\in A\mathit{\text{ for all permutations }}\pi \}.$$

Furthermore, the symmetric function

$$\overline{\psi }(\mathbf{x})=\sum _{1\le i_1<\cdots <i_k\le n}\phi (x_{i_1},\dots ,x_{i_k})$$

is also Schur harmonically convex (concave) on B.

2 Definitions and lemmas

In order to prove some further results, in this section we recall useful definitions and lemmas.

Definition 1 [1, 16]

Let $\mathbf{x}=(x_1,\dots ,x_n)$ and $\mathbf{y}=(y_1,\dots ,y_n)\in {\mathbb{R}}^n$.

• (i) We say y majorizes x (x is said to be majorized by y), denoted by $\mathbf{x}\prec \mathbf{y}$, if ${\sum }_{i=1}^k{x}_{[i]}\le {\sum }_{i=1}^k{y}_{[i]}$ for $k=1,2,\dots ,n-1$ and ${\sum }_{i=1}^n{x}_i={\sum }_{i=1}^n{y}_i$, where $x_{[1]}\ge \cdots \ge x_{[n]}$ and $y_{[1]}\ge \cdots \ge y_{[n]}$ are rearrangements of x and y in a descending order.

• (ii) Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$, a function $\phi :\mathrm{\Omega }\to \mathbb{R}$ is said to be a Schur-convex function on Ω if $\mathbf{x}\prec \mathbf{y}$ on Ω implies $\phi (\mathbf{x})\le$ $\phi (\mathbf{y})$. A function φ is said to be a Schur-concave function on Ω if and only if −φ is Schur-convex function on Ω.

Definition 2 [1, 16]

Let $\mathbf{x}=(x_1,\dots ,x_n)$ and $\mathbf{y}=(y_1,\dots ,y_n)\in {\mathbb{R}}^n$, $0\le \alpha \le 1$. A set $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is said to be a convex set if $\mathbf{x},\mathbf{y}\in \mathrm{\Omega }$ implies $\alpha \mathbf{x}+(1-\alpha )\mathbf{y}=(\alpha x_1+(1-\alpha )y_1,\dots ,\alpha x_n+(1-\alpha )y_n)\in \mathrm{\Omega }$.

Definition 3 [1, 16]

• (i) A set $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is called a symmetric set if $\mathbf{x}\in \mathrm{\Omega }$ implies $\mathbf{x}P\in \mathrm{\Omega }$ for every $n\times n$ permutation matrix P.

• (ii) A function $\phi :\mathrm{\Omega }\to \mathbb{R}$ is called symmetric if for every permutation matrix P, $\phi (\mathbf{x}P)=\phi (\mathbf{x})$ for all $\mathbf{x}\in \mathrm{\Omega }$.

Definition 4 Let $\mathrm{\Omega }\subset {\mathbb{R}}_{+}^n$, $\mathbf{x}=(x_1,\dots ,x_n)\in \mathrm{\Omega }$ and $\mathbf{y}=(y_1,\dots ,y_n)\in \mathrm{\Omega }$.

• (i) [[3], p.64] A set Ω is called a geometrically convex set if $(x_1^{\alpha }{y}_1^{\beta },\dots ,x_n^{\alpha }{y}_n^{\beta })\in \mathrm{\Omega }$ for all $\mathbf{x},\mathbf{y}\in \mathrm{\Omega }$ and $\alpha ,\beta \in [0,1]$ such that $\alpha +\beta =1$.

• (ii) [[3], p.107] A function $\phi :\mathrm{\Omega }\to {\mathbb{R}}_{+}$ is said to be a Schur geometrically convex function on Ω if $(log{x}_1,\dots ,log{x}_n)\prec (log{y}_1,\dots ,log{y}_n)$ on Ω implies $\phi (\mathbf{x})\le$ $\phi (\mathbf{y})$. A function φ is said to be a Schur geometrically concave function on Ω if and only if −φ is a Schur geometrically convex function.

Definition 5 [17]

Let $\mathrm{\Omega }\subset {\mathbb{R}}_{+}^n$.

• (i) A set Ω is said to be a harmonically convex set if $\frac{\mathbf{xy}}{\lambda \mathbf{x}+(1-\lambda )\mathbf{y}}\in \mathrm{\Omega }$ for every $\mathbf{x},\mathbf{y}\in \mathrm{\Omega }$ and $\lambda \in [0,1]$, where $\mathbf{xy}={\sum }_{i=1}^n{x}_i{y}_i$ and $\frac{1}{\mathbf{x}}=(\frac{1}{x_1},\dots ,\frac{1}{x_n})$.

• (ii) A function $\phi :\mathrm{\Omega }\to {\mathbb{R}}_{+}$ is said to be a Schur harmonically convex function on Ω if $\frac{1}{\mathbf{x}}\prec \frac{1}{\mathbf{y}}$ implies $\phi (\mathbf{x})\le \phi (\mathbf{y})$. 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.

Definition 6 [18]

Let $I\subset {\mathbb{R}}_{+}$, $\phi :I\to {\mathbb{R}}_{+}$ be continuous.

• (i) A function φ is said to be a GA convex (concave) function on I if

  $$\phi (\sqrt{xy})\le (\ge )\frac{\phi (x)+\phi (y)}{2}$$

  for all $x,y\in I$.

• (ii) A function φ is said to be a HA convex (concave) function on I if

  $$\phi \left(\frac{2xy}{x+y}\right)\le (\ge )\frac{\phi (x)+\phi (y)}{2}$$

for all $x,y\in I$.

Lemma 1 [[16], p.57]

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a symmetric convex set with a nonempty interior ${\mathrm{\Omega }}^0$. $\phi :\mathrm{\Omega }\to \mathbb{R}$ is continuous on Ω and differentiable on ${\mathrm{\Omega }}^0$. Then φ is a Schur-convex (Schur-concave) function if and only if φ is symmetric on Ω and

$$(x_1-x_2)(\frac{\partial \phi }{\partial x_1}-\frac{\partial \phi }{\partial x_2})\ge 0(\le 0)$$

(3)

holds for any $\mathbf{x}=(x_1,\dots ,x_n)\in {\mathrm{\Omega }}^0$.

Lemma 2 [[3], p.108]

Let $\mathrm{\Omega }\subset {\mathbb{R}}_{+}^n$ be a symmetric geometrically convex set with a nonempty interior ${\mathrm{\Omega }}^0$. Let $\phi :\mathrm{\Omega }\to {\mathbb{R}}_{+}$ be continuous on Ω and differentiable on ${\mathrm{\Omega }}^0$. Then φ is a Schur geometrically convex (Schur geometrically concave) function if and only if φ is symmetric on Ω and

$$(x_1-x_2)(x_1\frac{\partial \phi }{\partial x_1}-x_2\frac{\partial \phi }{\partial x_2})\ge 0(\le 0)$$

(4)

holds for any $\mathbf{x}=(x_1,\dots ,x_n)\in {\mathrm{\Omega }}^0$.

Lemma 3 [17, 19]

Let $\mathrm{\Omega }\subset {\mathbb{R}}_{+}^n$ be a symmetric harmonically convex set with a nonempty interior ${\mathrm{\Omega }}^0$. Let $\phi :\mathrm{\Omega }\to {\mathbb{R}}_{+}$ be continuous on Ω and differentiable on ${\mathrm{\Omega }}^0$. Then φ is a Schur harmonically convex (Schur harmonically concave) function if and only if φ is symmetric on Ω and

$$(x_1-x_2)(x_1^2\frac{\partial \phi }{\partial x_1}-x_2^2\frac{\partial \phi }{\partial x_2})\ge 0(\le 0)$$

(5)

holds for any $\mathbf{x}=(x_1,\dots ,x_n)\in {\mathrm{\Omega }}^0$.

Lemma 4 [18]

Let $I\subset {\mathbb{R}}_{+}$ be an open subinterval, and let $\phi :I\to {\mathbb{R}}_{+}$ be differentiable.

• (i) φ is GA-convex (concave) if and only if $x{\phi }^{\prime }(x)$ is increasing (decreasing).

• (ii) φ is HA-convex (concave) if and only if $x^2{\phi }^{\prime }(x)$ is increasing (decreasing).

3 Proofs of main results

Proof of Theorem 1 To verify condition (4) of Lemma 2, denote by ${\sum }_{\pi (i,j)}$ the summation over all permutations π such that $\pi (i)=1$, $\pi (j)=2$. Because φ is symmetric,

$$\begin{array}{r}\psi (x_1,\dots ,x_n)\\ =\underset{i\ne j}{\sum _{i,j\le k}}\sum _{\pi (i,j)}\phi (x_1,x_2,x_{\pi (1)},\dots ,x_{\pi (i-1)},x_{\pi (i+1)},\dots ,x_{\pi (j-1)},x_{\pi (j+1)},\dots ,x_{\pi (k)})\\ +\sum _{i\le k<j}\sum _{\pi (i,j)}\phi (x_1,x_{\pi (1)},\dots ,x_{\pi (i-1)},x_{\pi (i+1)},\dots ,x_{\pi (k)})\\ +\sum _{j\le k<i}\sum _{\pi (i,j)}\phi (x_2,x_{\pi (1)},\dots ,x_{\pi (j-1)},x_{\pi (j+1)},\dots ,x_{\pi (k)})\\ +\underset{i\ne j}{\sum _{k<i,j}}\sum _{\pi (i,j)}\phi (x_{\pi (1)},\dots ,x_{\pi (k)}).\end{array}$$

Then

$$\begin{array}{rl}{\mathrm{\Delta }}_1:=& (x_1\frac{\partial \psi }{\partial x_1}-x_2\frac{\partial \psi }{\partial x_2})(x_1-x_2)\\ =& \underset{i\ne j}{\sum _{i,j\le k}}\sum _{\pi (i,j)}[x_1{\phi }_{(1)}(x_1,x_2,x_{\pi (1)},\dots ,x_{\pi (i-1)},x_{\pi (i+1)},\dots ,x_{\pi (j-1)},x_{\pi (j+1)},\dots ,x_{\pi (k)})\\ -x_2{\phi }_{(2)}(x_1,x_2,x_{\pi (1)},\dots ,x_{\pi (i-1)},x_{\pi (i+1)},\dots ,x_{\pi (j-1)},x_{\pi (j+1)},\dots ,x_{\pi (k)})](x_1-x_2)\\ +\sum _{i\le k<j}\sum _{\pi (i,j)}[x_1{\phi }_{(1)}(x_1,x_{\pi (1)},\dots ,x_{\pi (i-1)},x_{\pi (i+1)},\dots ,x_{\pi (k)})\\ -x_2{\phi }_{(1)}(x_2,x_{\pi (1)},\dots ,x_{\pi (i-1)},x_{\pi (i+1)},\dots ,x_{\pi (k)})](x_1-x_2).\end{array}$$

Here,

$$(x_1{\phi }_{(1)}-x_2{\phi }_{(2)})(x_1-x_2)\ge 0(\le 0)$$

because φ is Schur geometrically convex (concave), and

$$[x_1{\phi }_{(1)}(x_1,z)-x_2{\phi }_{(1)}(x_2,z)](x_1-x_2)\ge 0(\le 0)$$

because $\phi (z,x_2,\dots ,x_k)$ is GA convex (concave) in its first argument on $\{z:(z,x_2,\dots ,x_k)\in A\}$. Accordingly, ${\mathrm{\Delta }}_1\ge 0$ (≤0). This shows that ψ is Schur geometrically convex (concave) on

$$B=\{(x_1,\dots ,x_n):(x_{\pi (1)},\dots ,x_{\pi (k)})\in A\mathit{\text{ for all permutations }}\pi \}.$$

Notice that

$$\overline{\psi }(\mathbf{x})=\psi (\mathbf{x})/k!(n-k)!.$$

Of course, $\overline{\psi }$ is Schur geometrically convex (concave) whenever ψ is Schur geometrically convex (concave).

The proof of Theorem 1 is completed. □

Proof of Theorem 2 We only need to verify condition (5) of Lemma 3, the proof is similar to that of Theorem 1 and is omitted. □

Remark 1 In most applications, A has the form $I^k$ for some interval $I\subset R$ and in this case $B=I^n$. Notice that the convexity of φ in its first argument also implies that φ is convex in each argument, the other arguments being fixed, because φ is symmetric.

4 Applications

Let

$$E_k\left(\frac{\mathbf{x}}{1-\mathbf{x}}\right)=\sum _{1\le i_1<\cdots <i_k\le n}\prod _{j=1}^k\frac{x_{i_j}}{1-x_{i_j}}.$$

(6)

In 2011, Guan and Guan [20] proved the following theorem through Lemma 2.

Theorem 3 The symmetric function $E_k(\frac{\mathbf{x}}{1-\mathbf{x}})$, $k=1,\dots ,n$, is Schur geometrically convex on ${(0,1)}^n$.

Now, we give a new proof of Theorem 3 by using Theorem 1. Furthermore, we prove the following theorem through Theorem 2.

Theorem 4 The symmetric function $E_k(\frac{\mathbf{x}}{1-\mathbf{x}})$, $k=1,\dots ,n$, is Schur harmonically convex on ${(0,1)}^n$.

Proof of Theorem 3 Let $\phi (\mathbf{z})={\prod }_{i=1}^k[z_i/(1-z_i)]$. Then

$$log\phi (\mathbf{z})=\sum _{i=1}^k[log{z}_i-log(1-z_i)]$$

and

$$\begin{array}{r}\frac{\partial \phi (\mathbf{z})}{\partial z_1}=\phi (\mathbf{z})(\frac{1}{z_1}+\frac{1}{1-z_1}),\frac{\partial \phi (\mathbf{z})}{\partial z_2}=\phi (\mathbf{z})(\frac{1}{z_2}+\frac{1}{1-z_2}),\\ \mathrm{\Delta }:=(z_1-z_2)(z_1\frac{\partial \phi (\mathbf{z})}{\partial z_1}-z_2\frac{\partial \phi (\mathbf{z})}{\partial z_2})\\ \phantom{\mathrm{\Delta }}=(z_1-z_2)\phi (\mathbf{z})(\frac{z_1}{1-z_1}-\frac{z_2}{1-z_2})\\ \phantom{\mathrm{\Delta }}={(z_1-z_2)}^2\phi (\mathbf{z})\frac{1}{(1-z_2)(1-z_1)}.\end{array}$$

(7)

This shows that $\mathrm{\Delta }\ge 0$ when $0<z_i<1$, $i=1,\dots ,k$. According to Lemma 2, φ is Schur geometrically convex on $A=\{\mathbf{z}:\mathbf{z}\in {(0,1)}^k\}$. Let $g(t)=\frac{t}{1-t}$, then $h(t):=t{g}^{\prime }(t)=\frac{t}{{(1-t)}^2}$. From $t\in (0,1)$, it follows that $h^{\prime }(t)=\frac{1+t}{{(1-t)}^3}\ge 0$. According to Lemma 4(i), φ is GA convex in its single variable on $(0,1)$. So $E_k(\frac{\mathbf{x}}{1-\mathbf{x}})$ is Schur geometrically convex on ${(0,1)}^n$ from Theorem 1. The proof of Theorem 3 is completed. □

Proof of Theorem 4 Let $\phi (\mathbf{z})={\prod }_{i=1}^k(z_i/1-z_i)$, then

$$log\phi (\mathbf{z})=\sum _{i=1}^k[log{z}_i-log(1-z_i)].$$

From (7), we get

$$\begin{array}{rl}{\mathrm{\Delta }}_1& :=(z_1-z_2)(z_1^2\frac{\partial \phi (\mathbf{z})}{\partial z_1}-z_2^2\frac{\partial \phi (\mathbf{z})}{\partial z_2})\\ =(z_1-z_2)\phi (\mathbf{z})(z_1-z_2+\frac{z_1^2}{1-z_1}-\frac{z_2^2}{1-z_2})\\ ={(z_1-z_2)}^2\phi (\mathbf{z})[1+\frac{z_1+z_2-z_1{z}_2}{(1-z_2)(1-z_1)}].\end{array}$$

This shows that ${\mathrm{\Delta }}_1\ge 0$ when $0<z_i<1$, $i=1,\dots ,k$. According to Lemma 3, φ is Schur harmonically convex on $A=\{\mathbf{z}:\mathbf{z}\in {(0,1)}^k\}$. Let $g(t)=\frac{t}{1-t}$, then $p(t):=t^2{g}^{\prime }(t)=\frac{t^2}{{(1-t)}^2}$. From $t\in (0,1)$, it follows that $p^{\prime }(t)=\frac{2t}{{(1-t)}^3}\ge 0$. According to Lemma 4(ii), φ is HA convex in its single variable on $(0,1)$. So $E_k(\frac{\mathbf{x}}{1-\mathbf{x}})$ is Schur harmonically convex on ${(0,1)}^n$ from Theorem 2. The proof of Theorem 4 is completed. □

By using Theorem A, the following conclusion is proved in [[1], p.129].

The symmetric function

$$\overline{\psi }(\mathbf{x})=\sum _{1\le i_1<\cdots <i_k\le n}\frac{x_{i_1}+\cdots +x_{i_k}}{x_{i_1}\cdots x_{i_k}}$$

(8)

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

Now we use Theorem 1 and Theorem 2, respectively, to study Schur geometric convexity and Schur harmonic convexity of $\overline{\psi }(\mathbf{x})$.

Theorem 5 The symmetric function $\overline{\psi }(\mathbf{x})$ is Schur geometrically convex and Schur harmonically concave on ${\mathbb{R}}_{+}^n$.

Proof Let $\phi (\mathbf{y})={\sum }_{i=1}^k{y}_i/{\prod }_{i=1}^k{y}_i$, then $log\phi (\mathbf{y})=log({\sum }_{i=1}^k{y}_i)-{\sum }_{i=1}^{k}log{y}_i$. Thus,

$$\begin{array}{r}\frac{\partial \phi (\mathbf{y})}{\partial y_1}=\phi (\mathbf{y})(\frac{1}{{\sum }_{i=1}^k{y}_i}-\frac{1}{y_1}),\frac{\partial \phi (\mathbf{y})}{\partial y_2}=\phi (\mathbf{y})(\frac{1}{{\sum }_{i=1}^k{y}_i}-\frac{1}{y_2}),\\ \mathrm{\Delta }:=(y_1-y_2)(y_1\frac{\partial \phi (\mathbf{y})}{\partial y_1}-y_2\frac{\partial \phi (\mathbf{y})}{\partial y_2})\\ \phantom{\mathrm{\Delta }}=(y_1-y_2)\phi (\mathbf{y})\left(\frac{y_1-y_2}{{\sum }_{i=1}^k{y}_i}\right)\\ \phantom{\mathrm{\Delta }}=\frac{{(y_1-y_2)}^2}{{\prod }_{i=1}^k{y}_i}\ge 0.\end{array}$$

According to Lemma 2, $\phi (\mathbf{y})$ is Schur geometrically convex on ${\mathbb{R}}_{+}^k$. Let $g(z)=\phi (z,x_2,\dots ,x_k)=\frac{z+a}{bz}=\frac{1}{b}+\frac{a}{bz}$, where $a={\sum }_{i=2}^k{x}_i$, $b={\prod }_{i=2}^k{x}_i$, then $h(z):=z{g}^{\prime }(z)=-\frac{a}{bz}$. From $z\in {\mathbb{R}}_{+}$, it follows that $h^{\prime }(z)=\frac{a}{b{z}^2}\ge 0$. According to Lemma 4(i), φ is GA convex in its single variable on ${\mathbb{R}}_{+}$. So $\overline{\psi }(\mathbf{x})$ is Schur geometrically convex on ${\mathbb{R}}_{+}$ from Theorem 1.

It is easy to check that

$$\begin{array}{rl}{\mathrm{\Delta }}_1& :=(y_1-y_2)(y_1^2\frac{\partial \phi (\mathbf{y})}{\partial y_1}-y_2^2\frac{\partial \phi (\mathbf{y})}{\partial y_2})\\ =\frac{{(y_1-y_2)}^2(y_1+y_2-{\sum }_{i=1}^k{y}_i)}{{\prod }_{i=1}^k{y}_i}\le 0.\end{array}$$

According to Lemma 3, $\phi (\mathbf{y})$ is Schur harmonically concave on ${\mathbb{R}}_{+}^k$. Let $h(z):=z^2{g}^{\prime }(z)=-\frac{a}{b}$. $h^{\prime }(z)=0$ when $z\in {\mathbb{R}}_{+}$. According to Lemma 4(ii), φ is HA concave in its single variable on ${\mathbb{R}}_{+}$. So $\overline{\psi }(\mathbf{x})$ is Schur harmonically concave on ${\mathbb{R}}_{+}^n$ from Theorem 2. □

Remark 2 Let

$$H=\frac{n}{{\sum }_{i=1}^n\frac{1}{x_i}},G={\left(\prod _{i=1}^n{x}_i\right)}^{\frac{1}{n}},$$

where $x_i>0$, $i=1,\dots ,n$. Then

$$(logG,\dots ,logG)\prec (log{x}_1,\dots ,log{x}_n),$$

(9)

$$(\frac{1}{H},\dots ,\frac{1}{H})\prec (\frac{1}{x_1},\dots ,\frac{1}{x_n}).$$

(10)

From Theorem 5, it follows that

$$\frac{k{C}_n^k}{H^{k-1}}\ge \sum _{1\le i_1<\cdots <i_k\le n}\frac{x_{i_1}+\cdots +x_{i_k}}{x_{i_1}\cdots x_{i_k}}\ge \frac{k{C}_n^k}{G^{k-1}}.$$

(11)

By using Theorem A, the following conclusion is proved in [[1], p.129].

The symmetric function

$$\psi (\mathbf{x})=\sum _{1\le i_1<\cdots <i_k\le n}\frac{x_{i_1}\cdots x_{i_k}}{x_{i_1}+\cdots +x_{i_k}}$$

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

By applying Theorem 2, we further obtain the following result.

Theorem 6 The symmetric function $\psi (\mathbf{x})$ is Schur harmonically convex on ${\mathbb{R}}_{+}^n$.

Proof Let $\lambda (\mathbf{y})={\prod }_{i=1}^k{y}_i/{\sum }_{i=1}^k{y}_i$. According to the proof of Theorem 5, $\phi (\mathbf{y})$ is Schur harmonically concave on ${\mathbb{R}}_{+}^k$. Let $\lambda (\mathbf{y})=\frac{1}{\phi (\mathbf{y})}$. From the definition of Schur harmonically convex, it follows that $\lambda (\mathbf{y})$ is Schur harmonically convex on ${\mathbb{R}}_{+}^k$. Let $g(z)=\lambda (z,x_2,\dots ,x_k)=\frac{bz}{z+a}$, where $a={\sum }_{i=2}^k{x}_i$, $b={\prod }_{i=2}^k{x}_i$. Then $h(z):=z^2{g}^{\prime }(z)=\frac{z^{2}ab}{{(z+a)}^2}$. With the fact that $h^{\prime }(z)=\frac{2z{a}^{2}b}{{(z+a)}^3}\ge 0$ for $z\in {\mathbb{R}}_{+}$, it follows that φ is HA convex in its single variable on ${\mathbb{R}}_{+}$. So, from Theorem 2, $\psi (\mathbf{x})$ is Schur harmonically convex on ${\mathbb{R}}_{+}^n$. □

Remark 3 From Theorem 6 and (10), it follows that

$$\sum _{1\le i_1<\cdots <i_k\le n}\frac{x_{i_1}\cdots x_{i_k}}{x_{i_1}+\cdots +x_{i_k}}\ge \frac{H^{k-1}{C}_n^k}{k},$$

(12)

where $x_i>0$, $i=1,\dots ,n$.

Remark 4 It needs further discussion that $\psi (\mathbf{x})$ is Schur geometrically convex on ${\mathbb{R}}_{+}^n$.