## 1 Introduction

Throughout this paper, ℝ denotes the set of real numbers, $\mathbf{x}=\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)$ denotes n-tuple (n-dimensional real vectors), the set of vectors can be written as

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

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

Let $\pi =\left(\pi \left(1\right),\dots ,\pi \left(n\right)\right)$ be a permutation of $\left(1,\dots ,n\right)$, 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 ${x}_{2},\dots ,{x}_{k}$, $\phi \left(z,{x}_{2},\dots ,{x}_{k}\right)$ is convex in z on $\left\{z:\left(z,{x}_{2},\dots ,{x}_{k}\right)\in A\right\}$. Then, for any $n>k$,

$\psi \left({x}_{1},\dots ,{x}_{n}\right)=\sum _{\pi }\phi \left({x}_{\pi \left(1\right)},\dots ,{x}_{\pi \left(k\right)}\right)$
(1)

is Schur-convex on

Furthermore, the symmetric function

$\overline{\psi }\left(\mathbf{x}\right)=\sum _{1\le {i}_{1}<\cdots <{i}_{k}\le n}\phi \left({x}_{{i}_{1}},\dots ,{x}_{{i}_{k}}\right)$
(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 [515]. 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 ${x}_{2},\dots ,{x}_{k}$, $\phi \left(z,{x}_{2},\dots ,{x}_{k}\right)$ is GA convex (concave) in z on $\left\{z:\left(z,{x}_{2},\dots ,{x}_{k}\right)\in A\right\}$. Then, for any $n>k$,

$\psi \left({x}_{1},\dots ,{x}_{n}\right)=\sum _{\pi }\phi \left({x}_{\pi \left(1\right)},\dots ,{x}_{\pi \left(k\right)}\right)$

is Schur geometrically convex (concave) on

Furthermore, the symmetric function

$\overline{\psi }\left(\mathbf{x}\right)=\sum _{1\le {i}_{1}<\cdots <{i}_{k}\le n}\phi \left({x}_{{i}_{1}},\dots ,{x}_{{i}_{k}}\right)$

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 ${x}_{2},\dots ,{x}_{k}$, $\phi \left(z,{x}_{2},\dots ,{x}_{k}\right)$ is HA convex (concave) in z on $\left\{z:\left(z,{x}_{2},\dots ,{x}_{k}\right)\in A\right\}$. Then, for any $n>k$,

$\psi \left({x}_{1},\dots ,{x}_{n}\right)=\sum _{\pi }\phi \left({x}_{\pi \left(1\right)},\dots ,{x}_{\pi \left(k\right)}\right)$

is Schur harmonically convex (concave) on

Furthermore, the symmetric function

$\overline{\psi }\left(\mathbf{x}\right)=\sum _{1\le {i}_{1}<\cdots <{i}_{k}\le n}\phi \left({x}_{{i}_{1}},\dots ,{x}_{{i}_{k}}\right)$

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 $\mathbf{x}=\left({x}_{1},\dots ,{x}_{n}\right)$ and $\mathbf{y}=\left({y}_{1},\dots ,{y}_{n}\right)\in {\mathbb{R}}^{n}$.

1. (i)

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

2. (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 $\phi \left(\mathbf{x}\right)\le$ $\phi \left(\mathbf{y}\right)$. 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 $\mathbf{x}=\left({x}_{1},\dots ,{x}_{n}\right)$ and $\mathbf{y}=\left({y}_{1},\dots ,{y}_{n}\right)\in {\mathbb{R}}^{n}$, $0\le \alpha \le 1$. A set $\mathrm{\Omega }\subset {\mathbb{R}}^{n}$ is said to be a convex set if $\mathbf{x},\mathbf{y}\in \mathrm{\Omega }$ implies $\alpha \mathbf{x}+\left(1-\alpha \right)\mathbf{y}=\left(\alpha {x}_{1}+\left(1-\alpha \right){y}_{1},\dots ,\alpha {x}_{n}+\left(1-\alpha \right){y}_{n}\right)\in \mathrm{\Omega }$.

Definition 3 [1, 16]

1. (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 $n×n$ permutation matrix P.

2. (ii)

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

Definition 4 Let $\mathrm{\Omega }\subset {\mathbb{R}}_{+}^{n}$, $\mathbf{x}=\left({x}_{1},\dots ,{x}_{n}\right)\in \mathrm{\Omega }$ and $\mathbf{y}=\left({y}_{1},\dots ,{y}_{n}\right)\in \mathrm{\Omega }$.

1. (i)

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

2. (ii)

[[3], p.107] A function $\phi :\mathrm{\Omega }\to {\mathbb{R}}_{+}$ is said to be a Schur geometrically convex function on Ω if $\left(log{x}_{1},\dots ,log{x}_{n}\right)\prec \left(log{y}_{1},\dots ,log{y}_{n}\right)$ on Ω implies $\phi \left(\mathbf{x}\right)\le$ $\phi \left(\mathbf{y}\right)$. 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 $\mathrm{\Omega }\subset {\mathbb{R}}_{+}^{n}$.

1. (i)

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

2. (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 $\phi \left(\mathbf{x}\right)\le \phi \left(\mathbf{y}\right)$. 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.

1. (i)

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

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

for all $x,y\in I$.

2. (ii)

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

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

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

$\left({x}_{1}-{x}_{2}\right)\left(\frac{\partial \phi }{\partial {x}_{1}}-\frac{\partial \phi }{\partial {x}_{2}}\right)\ge 0\phantom{\rule{0.25em}{0ex}}\left(\le 0\right)$
(3)

holds for any $\mathbf{x}=\left({x}_{1},\dots ,{x}_{n}\right)\in {\mathrm{\Omega }}^{0}$.

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

$\left({x}_{1}-{x}_{2}\right)\left({x}_{1}\frac{\partial \phi }{\partial {x}_{1}}-{x}_{2}\frac{\partial \phi }{\partial {x}_{2}}\right)\ge 0\phantom{\rule{0.25em}{0ex}}\left(\le 0\right)$
(4)

holds for any $\mathbf{x}=\left({x}_{1},\dots ,{x}_{n}\right)\in {\mathrm{\Omega }}^{0}$.

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

$\left({x}_{1}-{x}_{2}\right)\left({x}_{1}^{2}\frac{\partial \phi }{\partial {x}_{1}}-{x}_{2}^{2}\frac{\partial \phi }{\partial {x}_{2}}\right)\ge 0\phantom{\rule{0.25em}{0ex}}\left(\le 0\right)$
(5)

holds for any $\mathbf{x}=\left({x}_{1},\dots ,{x}_{n}\right)\in {\mathrm{\Omega }}^{0}$.

Lemma 4 [18]

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

1. (i)

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

2. (ii)

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

## 3 Proofs of main results

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

$\begin{array}{r}\psi \left({x}_{1},\dots ,{x}_{n}\right)\\ \phantom{\rule{1em}{0ex}}=\underset{i\ne j}{\sum _{i,j\le k}}\sum _{\pi \left(i,j\right)}\phi \left({x}_{1},{x}_{2},{x}_{\pi \left(1\right)},\dots ,{x}_{\pi \left(i-1\right)},{x}_{\pi \left(i+1\right)},\dots ,{x}_{\pi \left(j-1\right)},{x}_{\pi \left(j+1\right)},\dots ,{x}_{\pi \left(k\right)}\right)\\ \phantom{\rule{2em}{0ex}}+\sum _{i\le k

Then

$\begin{array}{rl}{\mathrm{\Delta }}_{1}:=& \left({x}_{1}\frac{\partial \psi }{\partial {x}_{1}}-{x}_{2}\frac{\partial \psi }{\partial {x}_{2}}\right)\left({x}_{1}-{x}_{2}\right)\\ =& \underset{i\ne j}{\sum _{i,j\le k}}\sum _{\pi \left(i,j\right)}\left[{x}_{1}{\phi }_{\left(1\right)}\left({x}_{1},{x}_{2},{x}_{\pi \left(1\right)},\dots ,{x}_{\pi \left(i-1\right)},{x}_{\pi \left(i+1\right)},\dots ,{x}_{\pi \left(j-1\right)},{x}_{\pi \left(j+1\right)},\dots ,{x}_{\pi \left(k\right)}\right)\\ -{x}_{2}{\phi }_{\left(2\right)}\left({x}_{1},{x}_{2},{x}_{\pi \left(1\right)},\dots ,{x}_{\pi \left(i-1\right)},{x}_{\pi \left(i+1\right)},\dots ,{x}_{\pi \left(j-1\right)},{x}_{\pi \left(j+1\right)},\dots ,{x}_{\pi \left(k\right)}\right)\right]\left({x}_{1}-{x}_{2}\right)\\ +\sum _{i\le k

Here,

$\left({x}_{1}{\phi }_{\left(1\right)}-{x}_{2}{\phi }_{\left(2\right)}\right)\left({x}_{1}-{x}_{2}\right)\ge 0\phantom{\rule{0.25em}{0ex}}\left(\le 0\right)$

because φ is Schur geometrically convex (concave), and

$\left[{x}_{1}{\phi }_{\left(1\right)}\left({x}_{1},z\right)-{x}_{2}{\phi }_{\left(1\right)}\left({x}_{2},z\right)\right]\left({x}_{1}-{x}_{2}\right)\ge 0\phantom{\rule{0.25em}{0ex}}\left(\le 0\right)$

because $\phi \left(z,{x}_{2},\dots ,{x}_{k}\right)$ is GA convex (concave) in its first argument on $\left\{z:\left(z,{x}_{2},\dots ,{x}_{k}\right)\in A\right\}$. Accordingly, ${\mathrm{\Delta }}_{1}\ge 0$ (≤0). This shows that ψ is Schur geometrically convex (concave) on

Notice that

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

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 ${E}_{k}\left(\frac{\mathbf{x}}{1-\mathbf{x}}\right)$, $k=1,\dots ,n$, is Schur geometrically convex on ${\left(0,1\right)}^{n}$.

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 ${E}_{k}\left(\frac{\mathbf{x}}{1-\mathbf{x}}\right)$, $k=1,\dots ,n$, is Schur harmonically convex on ${\left(0,1\right)}^{n}$.

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

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

and

$\begin{array}{r}\frac{\partial \phi \left(\mathbf{z}\right)}{\partial {z}_{1}}=\phi \left(\mathbf{z}\right)\left(\frac{1}{{z}_{1}}+\frac{1}{1-{z}_{1}}\right),\phantom{\rule{2em}{0ex}}\frac{\partial \phi \left(\mathbf{z}\right)}{\partial {z}_{2}}=\phi \left(\mathbf{z}\right)\left(\frac{1}{{z}_{2}}+\frac{1}{1-{z}_{2}}\right),\\ \mathrm{\Delta }:=\left({z}_{1}-{z}_{2}\right)\left({z}_{1}\frac{\partial \phi \left(\mathbf{z}\right)}{\partial {z}_{1}}-{z}_{2}\frac{\partial \phi \left(\mathbf{z}\right)}{\partial {z}_{2}}\right)\\ \phantom{\mathrm{\Delta }}=\left({z}_{1}-{z}_{2}\right)\phi \left(\mathbf{z}\right)\left(\frac{{z}_{1}}{1-{z}_{1}}-\frac{{z}_{2}}{1-{z}_{2}}\right)\\ \phantom{\mathrm{\Delta }}={\left({z}_{1}-{z}_{2}\right)}^{2}\phi \left(\mathbf{z}\right)\frac{1}{\left(1-{z}_{2}\right)\left(1-{z}_{1}\right)}.\end{array}$
(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 $A=\left\{\mathbf{z}:\mathbf{z}\in {\left(0,1\right)}^{k}\right\}$. Let $g\left(t\right)=\frac{t}{1-t}$, then $h\left(t\right):=t{g}^{\prime }\left(t\right)=\frac{t}{{\left(1-t\right)}^{2}}$. From $t\in \left(0,1\right)$, it follows that ${h}^{\prime }\left(t\right)=\frac{1+t}{{\left(1-t\right)}^{3}}\ge 0$. According to Lemma 4(i), φ is GA convex in its single variable on $\left(0,1\right)$. So ${E}_{k}\left(\frac{\mathbf{x}}{1-\mathbf{x}}\right)$ is Schur geometrically convex on ${\left(0,1\right)}^{n}$ from Theorem 1. The proof of Theorem 3 is completed. □

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

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

From (7), we get

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

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 $A=\left\{\mathbf{z}:\mathbf{z}\in {\left(0,1\right)}^{k}\right\}$. Let $g\left(t\right)=\frac{t}{1-t}$, then $p\left(t\right):={t}^{2}{g}^{\prime }\left(t\right)=\frac{{t}^{2}}{{\left(1-t\right)}^{2}}$. From $t\in \left(0,1\right)$, it follows that ${p}^{\prime }\left(t\right)=\frac{2t}{{\left(1-t\right)}^{3}}\ge 0$. According to Lemma 4(ii), φ is HA convex in its single variable on $\left(0,1\right)$. So ${E}_{k}\left(\frac{\mathbf{x}}{1-\mathbf{x}}\right)$ is Schur harmonically convex on ${\left(0,1\right)}^{n}$ 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

$\overline{\psi }\left(\mathbf{x}\right)=\sum _{1\le {i}_{1}<\cdots <{i}_{k}\le n}\frac{{x}_{{i}_{1}}+\cdots +{x}_{{i}_{k}}}{{x}_{{i}_{1}}\cdots {x}_{{i}_{k}}}$
(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 $\overline{\psi }\left(\mathbf{x}\right)$.

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

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

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

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

It is easy to check that

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

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

Remark 2 Let

$H=\frac{n}{{\sum }_{i=1}^{n}\frac{1}{{x}_{i}}},\phantom{\rule{2em}{0ex}}G={\left(\prod _{i=1}^{n}{x}_{i}\right)}^{\frac{1}{n}},$

where ${x}_{i}>0$, $i=1,\dots ,n$. Then

$\left(logG,\dots ,logG\right)\prec \left(log{x}_{1},\dots ,log{x}_{n}\right),$
(9)
$\left(\frac{1}{H},\dots ,\frac{1}{H}\right)\prec \left(\frac{1}{{x}_{1}},\dots ,\frac{1}{{x}_{n}}\right).$
(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

$\psi \left(\mathbf{x}\right)=\sum _{1\le {i}_{1}<\cdots <{i}_{k}\le n}\frac{{x}_{{i}_{1}}\cdots {x}_{{i}_{k}}}{{x}_{{i}_{1}}+\cdots +{x}_{{i}_{k}}}$

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

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

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

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