1 Introduction

The Schur convexity of functions relating to special means is a very significant research subject and has attracted the interest of many mathematicians. There are numerous articles written on this topic in recent years; see [1, 2] and the references therein. As supplements to the Schur convexity of functions, the Schur geometrically convex functions and Schur harmonically convex functions were investigated by Zhang and Yang [3], Chu, Zhang and Wang [4], Chu and Xia [5], Chu, Wang and Zhang [6], Shi and Zhang [7, 8], Meng, Chu and Tang [9], Zheng, Zhang and Zhang [10]. These properties of functions have been found to be useful in discovering and proving the inequalities for special means (see [11–14]).

Recently, it has come to our attention that a type of means which is symmetrical on n variables \(x_{1}, x_{2}, \ldots , x_{n}\) and involves two parameters, it was initially proposed by Bonferroni [15], as follows:

$$ B^{p,q}(\boldsymbol {x})= \Biggl( \frac{1}{n(n-1)}\sum ^{n}_{i,j=1,i\neq j} x^{p} _{i} x^{q}_{j} \Biggr) ^{\frac{1}{p+q}}, $$
(1)

where \(\boldsymbol {x}=(x_{1},x_{2},\ldots,x_{n})\), \(x_{i}\geq 0\), \(i=1,2,\ldots,n\), \(p,q \geq 0\) and \(p+q\neq 0\).

\(B^{p,q}(\boldsymbol {x})\) is called the Bonferroni mean. It has important application in multi criteria decision-making (see [16–21]).

Beliakov, James and Mordelová et al. [22] generalized the Bonferroni mean by introducing three parameters p, q, r, i.e.,

$$ B^{p,q,r}(\boldsymbol {x})= \Biggl( \frac{1}{n(n-1)(n-2)}\sum ^{n}_{i,j.k=1,i \neq j\neq k} x^{p}_{i} x^{q}_{j}x^{r}_{k} \Biggr) ^{\frac{1}{p+q+r}}, $$
(2)

where \(\boldsymbol {x}=(x_{1},x_{2},\ldots,x_{n})\), \(x_{i}\geq 0\), \(i=1,2,\ldots,n\), \(p,q,r \geq 0\) and \(p+q+r \neq 0\).

Motivated by the Bonferroni mean \(B^{p,q}(\boldsymbol {x})\) and the geometric mean \(G(\boldsymbol {x})=\prod^{n}_{i=1}(x_{i})^{\frac{1}{n}}\), Xia, Xu and Zhu [23] introduced a new mean which is called the geometric Bonferroni mean, as follows:

$$ \operatorname {GB}^{p,q}(\boldsymbol {x})=\frac{1}{p+q}\prod^{n}_{i,j=1,i\neq j}(p x_{i}+q x _{j})^{\frac{1}{n(n-1)}}, $$
(3)

where \(\boldsymbol {x}=(x_{1},x_{2},\ldots,x_{n})\), \(x_{i} > 0\), \(i=1,2,\ldots,n\), \(p,q \geq 0\) and \(p+q\neq 0\).

An extension of the geometric Bonferroni mean was given by Park and Kim in [19], which is called the generalized geometric Bonferroni mean, i.e.,

$$ \operatorname {GB}^{p,q,r}(\boldsymbol {x})=\frac{1}{p+q+r}\prod^{n}_{i,j,k=1,i\neq j\neq k}(p x_{i}+q x_{j}+r x_{k})^{\frac{1}{n(n-1)(n-2)}}, $$
(4)

where \(\boldsymbol {x}=(x_{1},x_{2},\ldots,x_{n})\), \(x_{i}> 0\), \(i=1,2,\ldots,n\), \(p,q,r \geq 0\) and \(p+q+r \neq 0\).

Remark 1

For \(r=0\), it is easy to observe that

$$\begin{aligned} \operatorname {GB}^{p,q,0}(\boldsymbol {x})&=\frac{1}{p+q+0}\prod ^{n}_{i,j=1,i\neq j} \Biggl[\prod^{n}_{k=1,i\neq j\neq k}(p x_{i}+q x_{j}+0\times x_{k}) \Biggr]^{ \frac{1}{n(n-1)(n-2)}} \\ & =\frac{1}{p+q}\prod^{n}_{i,j=1,i\neq j} \bigl[(p x_{i}+q x_{j})^{(n-2)} \bigr]^{\frac{1}{n(n-1)(n-2)}} \\ & =\frac{1}{p+q}\prod^{n}_{i,j=1,i\neq j}(p x_{i}+q x_{j})^{ \frac{1}{n(n-1)}} \\ & =\operatorname {GB}^{p,q}(\boldsymbol {x}). \end{aligned}$$

Remark 2

If \(q = 0\), \(r=0\), then the generalized geometric Bonferroni mean reduces to the geometric mean, i.e.,

$$ \operatorname {GB}^{p,0,0}(\boldsymbol {x})=\operatorname {GB}^{p,0}(\boldsymbol {x})=\frac{1}{p}\prod ^{n}_{i,j=1,i \neq j}(p x_{i})^{\frac{1}{n(n-1)}}= \prod^{n}_{i=1}(x_{i})^{ \frac{1}{n}}=G( \boldsymbol {x}). $$

Remark 3

If \(\boldsymbol {x}=(x,x,\ldots,x)\), then

$$ \operatorname {GB}^{p,q,r}(\boldsymbol {x})=\operatorname {GB}^{p,q,r}(x,x,\ldots,x)=x. $$

For convenience, throughout the paper \(\mathbb{R}\) 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

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

In a recent paper [24], Shi and Wu investigated the Schur m-power convexity of the geometric Bonferroni mean \(\operatorname {GB}^{p,q}(\boldsymbol {x})\). The definition of Schur m-power convex function is as follows:

Let \(f:\mathbb{R}_{++}\to \mathbb{R}\) be a function defined by

$$ f(x)= \textstyle\begin{cases} \frac{x^{m}-1}{m}, & m\ne 0, \\ \ln x, & m=0. \end{cases} $$

Then a function \(\varphi :\Omega \subset \mathbb{R}_{++}^{n}\to \mathbb{R}\) is said to be Schur m-power convex on Ω if

$$ \bigl(f(x_{1}), f(x_{2}),\ldots,f(x_{n}) \bigr) \prec \bigl(f(y_{1}), f(y_{2}),\ldots,f(y_{n}) \bigr) $$

for all \((x_{1},x_{2},\ldots,x_{n})\in \Omega \) and \((y_{1},y_{2},\ldots,y_{n})\in \Omega \) implies \(\phi (x)\le \phi (y)\).

If −φ is Schur m-power convex, then we say that φ is Schur m-power concave.

Shi and Wu [24] obtained the following result.

Proposition 1

For fixed positive real numbers p, q, (i) if \(m<0 \) or \(m=0 \), then \(\operatorname {GB}^{p,q}(\boldsymbol {x})\) is Schur m-power convex on \(\mathbb{R}^{n}_{++}\); (ii) if \(m=1 \) or \(m\geq 2 \), then \(\operatorname {GB}^{p,q}(\boldsymbol {x})\) is Schur m-power concave on \(\mathbb{R}^{n}_{++}\).

In this paper we discuss the Schur convexity, Schur geometric convexity and Schur harmonic convexity of the generalized geometric Bonferroni mean \(\operatorname {GB}^{p,q,r}(\boldsymbol {x})\). Our main results are as follows.

Theorem 1

For fixed non-negative real numbers p, q, r with \(p+q+r \neq 0\), if \(\boldsymbol {x}=(x_{1},x_{2},\ldots,x_{n})\), \(n\geq 3 \), then \(\operatorname {GB}^{p,q,r}(\boldsymbol {x})\) is Schur concave, Schur geometric convex and Schur harmonic convex on \(\mathbb{R}^{n}_{++}\).

Corollary 1

For fixed non-negative real numbers p, q with \(p+q\neq 0\), if \(\boldsymbol {x}=(x_{1},x_{2},\ldots,x_{n})\), \(n\geq 3 \), then \(\operatorname {GB}^{p,q}(\boldsymbol {x})\) is Schur concave, Schur geometric convex and Schur harmonic convex on \(\mathbb{R}^{n}_{++}\).

2 Preliminaries

We introduce some definitions, lemmas and propositions, which will be used in the proofs of the main results in subsequent sections.

Definition 1

(see [1])

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

  1. (i)

    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. (ii)

    Let \(\Omega \subset \mathbb{R}^{n}\),the function φ: \(\Omega \to \mathbb{R}\) is said to be Schur convex on Ω if \(\boldsymbol {x} \prec \boldsymbol {y}\) on Ω implies \(\varphi ( \boldsymbol {x} ) \le \varphi ( \boldsymbol {y} ) \). φ is said to be Schur concave function on Ω if and only if −φ is Schur convex function on Ω.

Definition 2

(see [1])

Let \(\boldsymbol {x} = ( x_{1},x_{2},\ldots, x_{n })\) and \(\boldsymbol {y} = ( y_{1},y_{2},\ldots, y_{n }) \in \mathbb{R}^{n}\). \(\Omega \subset \mathbb{R}^{n}\) is said to be a convex set if \(\boldsymbol {x},\boldsymbol {y}\in \Omega \) and \(0\leq \alpha \leq 1\) imply

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

Definition 3

(see [1])

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

(ii) A function \(\varphi : \Omega \to \mathbb{R}\) is called symmetric if for every permutation matrix P and \(\varphi (\boldsymbol {x}P) = \varphi (\boldsymbol {x})\) for all \(\boldsymbol {x} \in \Omega \).

The following proposition is called Schur’s condition. It provides an approach for testing whether a vector valued function is Schur convex or not.

Proposition 2

(see [1])

Let \(\Omega \subset \mathbb{R} ^{n} \) be symmetric and have a nonempty interior convex set. \(\Omega^{0}\) is the interior of Ω. \(\varphi :\Omega \to \mathbb{R} \) is continuous on Ω and differentiable in \(\Omega^{0}\). Then φ is the Schur convex function (Schur concave function) if and only if φ is symmetric on Ω and

$$ ( x_{1} - x_{2} ) \biggl[ \frac{\partial \varphi ( \boldsymbol{x})}{\partial x_{1}} - \frac{\partial \varphi ( \boldsymbol{x})}{\partial x_{2} } \biggr] \ge 0\quad (\leq 0) $$
(5)

holds for any \(\boldsymbol {x} \in \Omega^{0} \).

Definition 4

(see [25])

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. (i)

    \(\Omega \subset \mathbb{R}_{+} ^{n}\) is called a geometrically convex set if \((x_{1}^{\alpha }y_{1}^{\beta }, x_{2} ^{\alpha }y_{2}^{\beta },\ldots,x_{n}^{\alpha }y_{n}^{\beta }) \in \Omega \) for all x, \(\boldsymbol {y} \in \Omega \) and α, \(\beta \in [0, 1]\) such that \(\alpha +\beta =1\).

  2. (ii)

    Let \(\Omega \subset \mathbb{R}_{+} ^{n}\). The function φ: \(\Omega \to \mathbb{R}_{+}\) is said to be 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 Schur geometrically convex function.

Proposition 3

(see [25])

Let \(\Omega \subset \mathbb{R}_{+} ^{n} \) be a symmetric and geometrically convex set with a nonempty interior \(\Omega^{0}\). Let \(\varphi :\Omega \to \mathbb{R}_{+}\) be continuous on Ω and differentiable in \(\Omega^{0}\). If φ is symmetric on Ω and

$$ ( {\log x_{1} -\log x_{2} } ) \biggl[ {x_{1} \frac{\partial \varphi (\boldsymbol {x})}{\partial x_{1} } - x_{2} \frac{\partial \varphi (\boldsymbol {x})}{ \partial x_{2} }} \biggr] \ge 0 \quad ( \leq 0) $$
(6)

holds for any \(\boldsymbol {x}\in \Omega^{0} \), then φ is a Schur geometrically convex (Schur geometrically concave) function.

Definition 5

(see [26])

Let \(\Omega \subset \mathbb{R}_{+}^{n}\).

  1. (i)

    A set Ω is said to be harmonically convex if \(\frac{\boldsymbol{xy}}{\lambda {\boldsymbol{x}}+(1-\lambda ){\boldsymbol{y}}} \in \Omega \) for every \({\boldsymbol{x},\boldsymbol{y}}\in \Omega \) and \(\lambda \in [0,1]\), where \(\boldsymbol{xy}=(x_{1}y_{1},x_{2}y_{2},\ldots, x_{n}y_{n})\) and

    $$\frac{1}{\boldsymbol{\lambda x+(1-\lambda )y}} = \biggl( \frac{1}{\lambda x_{1}+(1-\lambda )y_{1}}, \frac{1}{\lambda x_{2}+(1- \lambda )y_{2}},\ldots, \frac{1}{\lambda x_{n}+(1-\lambda )y_{n}} \biggr). $$
  2. (ii)

    A function \(\varphi :\Omega \to \mathbb{R}_{+}\) 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.

Proposition 4

(see [26])

Let \(\Omega \subset \mathbb{R}_{+}^{n}\) be a symmetric and harmonically convex set with inner points, and let \(\varphi :\Omega \to \mathbb{R}_{+}\) be a continuously symmetric function which is differentiable on \(\Omega^{0}\). Then φ is Schur harmonically convex (Schur harmonically concave) on Ω if and only if

$$ (x_{1}-x_{2}) \biggl[ x_{1}^{2} \frac{\partial \varphi ({\boldsymbol{x}})}{ \partial x_{1}} -x_{2}^{2} \frac{\partial \varphi ({\boldsymbol{x}})}{ \partial x_{2}} \biggr] \ge 0\quad (\leq 0) $$
(7)

holds for any \(\boldsymbol {x} \in \Omega^{0} \).

Remark 4

Propositions 3 and 4 provide analogous Schur’s conditions for determining Schur geometrically convex functions and Schur harmonically convex functions, respectively.

Lemma 1

(see [1])

Let \(\boldsymbol {x}=( x_{1}, x_{2},\ldots, x_{n} )\in \mathbb{R}^{n}_{+} \) and \(A_{n}(\boldsymbol {x})= \frac{1}{n}\sum_{i=1}^{n}x_{i}\). Then

$$ \bigl( \underbrace{ A_{n}(\boldsymbol {x}), A_{n}(\boldsymbol {x}),\ldots, A_{n}(\boldsymbol {x})}_{n} \bigr) \prec (x_{1}, x_{2},\ldots, x_{n}). $$
(8)

Lemma 2

(see [1])

If \(x_{i}>0\), \(i=1,2,\ldots,n\), then, for any non-negative constant c satisfying \(0\leq c<\frac{1}{n}\sum_{i=1}^{n}x_{i}\), one has

$$ \biggl( \frac{x_{1}}{\sum_{i=1}^{n}x_{i}},\ldots,\frac{x_{n}}{\sum_{i=1} ^{n}x_{i}} \biggr) \prec \biggl( \frac{x_{1}-c}{\sum_{i=1}^{n}(x_{i}-c)},\ldots,\frac{x_{n}-c}{\sum_{i=1}^{n}(x_{i}-c)} \biggr) . $$
(9)

3 Proof of main result

Proof of Theorem 1

Note that the generalized geometric Bonferroni mean is defined by

$$ \operatorname {GB}^{p,q,r}(\boldsymbol {x})=\frac{1}{p+q+r}\prod^{n}_{i,j,k=1,i\neq j\neq k}(p x_{i}+q x_{j}+r x_{k})^{\frac{1}{n(n-1)(n-2)}}, $$

taking the natural logarithm gives

$$ \log \operatorname {GB}^{p,q,r}(\boldsymbol {x}) = \log \frac{1}{p+q+r}+\frac{1}{n(n-1)(n-2)}Q, $$

where

$$\begin{aligned} Q &= \sum^{n}_{j,k=3,j \neq k} \bigl[ \log (p x_{1}+q x_{j}+r x_{k})+ \log (p x_{2}+q x_{j}+r x_{k}) \bigr] \\ &\quad{} +\sum^{n}_{i,k=3,i \neq k} \bigl[ \log (p x_{i}+q x_{1}+r x_{k})+ \log (p x_{i}+q x_{2}+r x_{k}) \bigr] \\ &\quad{}+\sum^{n}_{i,j=3,i \neq j} \bigl[ \log (p x_{i}+q x_{j}+r x_{1})+ \log (p x_{i}+q x_{j}+r x_{2}) \bigr] \\ &\quad{}+\sum^{n}_{k=3} \bigl[\log (px_{1}+qx_{2}+ r x_{k})+\log (px_{2}+q x_{1}+ r x_{k}) \bigr] \\ &\quad{}+\sum^{n}_{j=3} \bigl[\log (px_{1}+qx_{j}+ r x_{2})+\log (px_{2}+q x_{j}+ r x_{1}) \bigr] \\ &\quad{}+\sum^{n}_{i=3} \bigl[\log (px_{i}+qx_{1}+ r x_{2})+\log (px_{i}+q x_{2}+ r x_{1}) \bigr] \\ &\quad{}+\sum^{n}_{i,j,k=3, i\neq j\neq k}\log (p x_{i}+q x_{j}+rx_{k}). \end{aligned}$$

Differentiating \(\operatorname {GB}^{p,q,r}(\boldsymbol {x})\) with respect to \(x_{1}\) and \(x_{2}\), respectively, we have

$$\begin{aligned}& \begin{aligned} \frac{\partial \operatorname {GB}^{p,q,r}(\boldsymbol {x})}{\partial x_{1}} &= \frac{\operatorname {GB}^{p,q,r}(\boldsymbol {x})}{n(n-1)(n-2)}\cdot \frac{\partial Q}{\partial x_{1}} \\ &=\frac{\operatorname {GB}^{p,q,r}(\boldsymbol {x})}{n(n-1)(n-2)} \Biggl[\sum^{n}_{j,k=3,j \neq k} \frac{p}{p x_{1}+q x_{j}+r x_{k}} +\sum^{n}_{i,k=3,i \neq k} \frac{q}{p x_{i}+q x_{1}+r x_{k}} \\ &\quad{}+\sum^{n}_{i,j=3,i \neq j}\frac{r}{p x_{i}+q x_{j}+r x_{1}} + \sum^{n}_{k=3} \biggl(\frac{p}{px_{1}+qx_{2}+ r x_{k}}+ \frac{q}{px_{2}+q x_{1}+ r x_{k}} \biggr) \\ &\quad{}+\sum^{n}_{j=3} \biggl( \frac{p}{px_{1}+qx_{j}+ r x_{2}}+ \frac{r}{px_{2}+q x _{j}+ r x_{1}} \biggr) \\ &\quad{} +\sum^{n}_{i=3} \biggl( \frac{q}{px_{i}+qx_{1}+ r x_{2}}+\frac{r}{px _{i}+q x_{2}+ r x_{1}} \biggr) \Biggr], \end{aligned} \\ & \begin{aligned} \frac{\partial \operatorname {GB}^{p,q,r}(\boldsymbol {x})}{\partial x_{2}}&= \frac{\operatorname {GB}^{p,q,r}(\boldsymbol {x})}{n(n-1)(n-2)}\cdot \frac{\partial Q}{\partial x_{2}} \\ &=\frac{\operatorname {GB}^{p,q,r}(\boldsymbol {x})}{n(n-1)(n-2)} \Biggl[\sum^{n}_{j,k=3,j \neq k} \frac{p}{p x_{2}+q x_{j}+r x_{k}} +\sum^{n}_{i,k=3,i \neq k} \frac{q}{p x_{i}+q x_{2}+r x_{k}} \\ &\quad{} +\sum^{n}_{i,j=3,i \neq j}\frac{r}{p x_{i}+q x_{j}+r x_{2}} +\sum^{n}_{k=3} \biggl(\frac{q}{px_{1}+qx_{2}+ r x_{k}}+ \frac{p}{px_{2}+q x_{1}+ r x_{k}} \biggr) \\ &\quad{} +\sum^{n}_{j=3} \biggl( \frac{r}{px_{1}+qx_{j}+ r x_{2}}+ \frac{p}{px_{2}+q x _{j}+ r x_{1}} \biggr) \\ &\quad{} +\sum^{n}_{i=3} \biggl( \frac{r}{px_{i}+qx_{1}+ r x_{2}}+\frac{q}{px _{i}+q x_{2}+ r x_{1}} \biggr) \Biggr]. \end{aligned} \end{aligned}$$

It is easy to see that \(\operatorname {GB}^{p,q,r}(\boldsymbol {x})\) is symmetric on \(\mathbb{R}^{n}_{++}\). For \(n\geq 3 \), we have

$$\begin{aligned} \Delta_{1}: ={}&(x_{1} -x_{2}) \biggl[ \frac{\partial \operatorname {GB}^{p,q,r}(\boldsymbol {x})}{ \partial x_{1}}-\frac{\partial \operatorname {GB}^{p,q,r}(\boldsymbol {x})}{\partial x_{2}} \biggr] \\ ={} &\frac{(x_{1} -x_{2})\operatorname {GB}^{p,q,r}(\boldsymbol {x})}{n(n-1)(n-2)} \Biggl[p \sum^{n}_{j,k=3,j \neq k} \biggl(\frac{1}{p x_{1}+q x_{j}+r x_{k}}-\frac{1}{p x _{2}+q x_{j}+r x_{k}} \biggr) \\ &{} +q\sum^{n}_{i,k=3,i \neq k} \biggl( \frac{1}{p x_{i}+q x_{1}+r x_{k}}- \frac{1}{p x_{i}+q x_{2}+r x_{k}} \biggr) \\ &{}+r\sum^{n}_{i,j=3,i \neq j} \biggl( \frac{1}{p x_{i}+q x_{j}+r x_{1}}- \frac{1}{p x_{i}+q x_{j}+r x_{2}} \biggr) \\ &{}+\sum^{n}_{k=3} \biggl( \frac{p-q}{px_{1}+qx_{2}+ r x_{k}}+ \frac{q-p}{px _{2}+q x_{1}+ r x_{k}} \biggr) \\ &{}+\sum^{n}_{j=3} \biggl( \frac{p-r}{px_{1}+qx_{j}+ r x_{2}}+ \frac{r-p}{px _{2}+q x_{j}+ r x_{1}} \biggr) \\ &{}+\sum^{n}_{i=3} \biggl( \frac{q-r}{px_{i}+qx_{1}+ r x_{2}}+ \frac{r-q}{px _{i}+q x_{2}+ r x_{1}} \biggr) \Biggr] \\ ={}&-\frac{(x_{1} -x_{2})^{2}\operatorname {GB}^{p,q,r}(\boldsymbol {x})}{n(n-1)(n-2)} \Biggl[ \sum^{n}_{j,k=3,j \neq k} \frac{p^{2}}{(p x_{1}+q x_{j}+r x_{k})(p x _{2}+q x_{j}+r x_{k})} \\ &{}+\sum^{n}_{i,k=3,i \neq k}\frac{q^{2}}{(p x_{i}+q x_{1}+r x_{k})(p x _{i}+q x_{2}+r x_{k})} \\ &{}+\sum^{n}_{i,j=3,i \neq j}\frac{r^{2}}{(p x_{i}+q x_{j}+r x_{1})(p x _{i}+q x_{j}+r x_{2})} \\ &{}+\sum^{n}_{k=3}\frac{(p-q)^{2}}{(px_{1}+qx_{2}+ r x_{k})(px_{2}+q x _{1}+ r x_{k})} \\ &{}+\sum^{n}_{j=3}\frac{(p-r)^{2}}{(px_{1}+qx_{j}+ r x_{2})(px_{2}+q x _{j}+ r x_{1})} \\ &{}+\sum^{n}_{i=3}\frac{(q-r)^{2}}{(px_{i}+qx_{1}+ r x_{2})(px_{i}+q x _{2}+ r x_{1})} \Biggr]. \end{aligned}$$

This implies that \(\Delta_{1}\leq 0\) for \(\boldsymbol {x}\in \mathbb{R}^{n} _{++}\) (\(n\geq 3\)). By Proposition 2, we conclude that \(\operatorname {GB}^{p,q,r}(\boldsymbol {x})\) is Schur concave on \(\mathbb{R}^{n}_{++}\).

In view of the discrimination criterion of Schur geometrically convexity, we start with the following calculations:

$$\begin{aligned} \Delta_{2}: ={}&(\log x_{1} - \log x_{2}) \biggl[ x_{1}\frac{\partial \operatorname {GB}^{p,q,r}(\boldsymbol {x})}{\partial x_{1}}-x_{2}\frac{\partial \operatorname {GB}^{p,q,r}(\boldsymbol {x})}{ \partial x_{2}} \biggr] \\ = {}&\frac{(\log x_{1} - \log x_{2})\operatorname {GB}^{p,q,r}(\boldsymbol {x})}{n(n-1)(n-2)} \\ &{}\times \Biggl[p \sum^{n}_{j,k=3,j \neq k} \biggl( \frac{x_{1}}{p x_{1}+q x _{j}+r x_{k}}-\frac{x_{2}}{p x_{2}+q x_{j}+r x_{k}} \biggr) \\ &{}+q\sum^{n}_{i,k=3,i \neq k} \biggl( \frac{x_{1}}{p x_{i}+q x_{1}+r x_{k}}- \frac{x _{2}}{p x_{i}+q x_{2}+r x_{k}} \biggr) \\ &{}+r\sum^{n}_{i,j=3,i \neq j} \biggl( \frac{x_{1}}{p x_{i}+q x_{j}+r x_{1}}- \frac{x _{2}}{p x_{i}+q x_{j}+r x_{2}} \biggr) \\ &{}+\sum^{n}_{k=3} \biggl( \frac{px_{1}-qx_{2}}{px_{1}+qx_{2}+ r x_{k}}+ \frac{qx _{1}-px_{2}}{px_{2}+q x_{1}+ r x_{k}} \biggr) \\ &{}+\sum^{n}_{j=3} \biggl( \frac{px_{1}-rx_{2}}{px_{1}+qx_{j}+ r x_{2}}+ \frac{rx _{1}-px_{2}}{px_{2}+q x_{j}+ r x_{1}} \biggr) \\ &{}+\sum^{n}_{i=3} \biggl( \frac{qx_{1}-rx_{2}}{px_{i}+qx_{1}+ r x_{2}}+ \frac{rx _{1}-qx_{2}}{px_{i}+q x_{2}+ r x_{1}} \biggr) \Biggr] \\ ={} &\frac{(x_{1} -x_{2})(\log x_{1} - \log x_{2})\operatorname {GB}^{p,q,r}(\boldsymbol {x})}{n(n-1)(n-2)} \\ &{}\times \Biggl[\sum^{n}_{j,k=3,j \neq k} \frac{q x_{j}+r x_{k}}{(p x _{1}+q x_{j}+r x_{k})(p x_{2}+q x_{j}+r x_{k})} \\ &{}+\sum^{n}_{i,k=3,i \neq k}\frac{px_{i}+r x_{k}}{(p x_{i}+q x_{1}+r x _{k})(p x_{i}+q x_{2}+r x_{k})} \\ &{}+\sum^{n}_{i,j=3,i \neq j}\frac{px_{i}+q x_{j}}{(p x_{i}+q x_{j}+r x _{1})(p x_{i}+q x_{j}+r x_{2})} \\ &{}+\sum^{n}_{k=3}\frac{2pq(x_{1}+x_{2})+rx_{k}(p+q)}{(px_{1}+qx_{2}+ r x_{k})(px_{2}+q x_{1}+ r x_{k})} \\ &{}+\sum^{n}_{j=3}\frac{2rp(x_{1}+x_{2})+qx_{j}(p+r)}{(px_{1}+qx_{j}+ r x_{2})(px_{2}+q x_{j}+ r x_{1})} \\ &{}+\sum^{n}_{i=3}\frac{2qr(x_{1}+x_{2})+px_{i}(q+r)}{(px_{i}+qx_{1}+ r x_{2})(px_{i}+q x_{2}+ r x_{1})} \Biggr]. \end{aligned}$$

Thus, we have \(\Delta_{2} \geq 0\) for \(\boldsymbol {x}\in \mathbb{R}^{n}_{++} \) (\(n\geq 3\)). It follows from Proposition 3 that \(\operatorname {GB}^{p,q,r}(\boldsymbol {x})\) is Schur geometric convex on \(\mathbb{R}^{n}_{++}\).

Finally, we discuss the Schur harmonic convexity of \(\operatorname {GB}^{p,q,r}(\boldsymbol {x})\). A direct computation gives

$$\begin{aligned} \Delta_{3}: ={}&(x_{1} -x_{2}) \biggl( x^{2}_{1}\frac{\partial \operatorname {GB}^{p,q,r}(\boldsymbol {x})}{ \partial x_{1}}-x^{2}_{2} \frac{\partial \operatorname {GB}^{p,q,r}(\boldsymbol {x})}{\partial x_{2}} \biggr) \\ = {}&\frac{(x_{1} -x_{2})\operatorname {GB}^{p,q,r}(\boldsymbol {x})}{n(n-1)(n-2)} \Biggl[p \sum^{n}_{j,k=3,j \neq k} \biggl(\frac{x^{2}_{1}}{p x_{1}+q x_{j}+r x_{k}}-\frac{x ^{2}_{2}}{p x_{2}+q x_{j}+r x_{k}} \biggr) \\ &{}+q\sum^{n}_{i,k=3,i \neq k} \biggl( \frac{x^{2}_{1}}{p x_{i}+q x_{1}+r x _{k}}- \frac{x^{2}_{2}}{p x_{i}+q x_{2}+r x_{k}} \biggr) \\ &{}+r\sum^{n}_{i,j=3,i \neq j} \biggl( \frac{x^{2}_{1}}{p x_{i}+q x_{j}+r x _{1}}- \frac{x^{2}_{2}}{p x_{i}+q x_{j}+r x_{2}} \biggr) \\ &{}+\sum^{n}_{k=3} \biggl( \frac{px^{2}_{1}-qx^{2}_{2}}{px_{1}+qx_{2}+ r x_{k}}+ \frac{qx^{2}_{1}-px ^{2}_{2}}{px_{2}+q x_{1}+ r x_{k}} \biggr) \\ &{}+\sum^{n}_{j=3} \biggl( \frac{px^{2}_{1}-rx^{2}_{2}}{px_{1}+qx_{j}+ r x_{2}}+\frac{rx^{2}_{1}-px ^{2}_{2}}{px_{2}+q x_{j}+ r x_{1}} \biggr) \\ &{}+\sum^{n}_{i=3} \biggl( \frac{qx^{2}_{1}-rx^{2}_{2}}{px_{i}+qx_{1}+ r x_{2}}+\frac{rx^{2}_{1}-qx ^{2}_{2}}{px_{i}+q x_{2}+ r x_{1}} \biggr) \Biggr] \\ ={}&\frac{(x_{1} -x_{2})^{2}\operatorname {GB}^{p,q,r}(\boldsymbol {x})}{n(n-1)(n-2)} \Biggl[\sum^{n}_{j,k=3,j \neq k} \frac{(x_{1}+x_{2})(q x_{j}+r x_{k})+px_{1}x_{2}}{(p x_{1}+q x_{j}+r x_{k})(p x_{2}+q x_{j}+r x_{k})} \\ &{}+\sum^{n}_{i,k=3,i \neq k}\frac{(x_{1}+x_{2})(px_{i}+r x_{k})+qx _{1}x_{2}}{(p x_{i}+q x_{1}+r x_{k})(p x_{i}+q x_{2}+r x_{k})} \\ &{}+\sum^{n}_{i,j=3,i \neq j}\frac{(x_{1}+x_{2})(px_{i}+q x_{j})+rx _{1}x_{2}}{(p x_{i}+q x_{j}+r x_{1})(p x_{i}+q x_{j}+r x_{2})} \\ &{}+\sum^{n}_{k=3}\frac{2pq(x^{2}_{1}+x^{2}_{2})+rx_{k}(x_{1}+x_{2})(p+q)+x _{1}x_{2}(p+q)^{2}}{(px_{1}+qx_{2}+ r x_{k})(px_{2}+q x_{1}+ r x_{k})} \\ &{}+\sum^{n}_{j=3}\frac{2pr(x^{2}_{1}+x^{2}_{2})+qx_{j}(x_{1}+x_{2})(p+r)+x _{1}x_{2}(p+r)^{2}}{(px_{1}+qx_{j}+ r x_{2})(px_{2}+q x_{j}+ r x_{1})} \\ &{}+\sum^{n}_{i=3}\frac{2qr(x^{2}_{1}+x^{2}_{2})+px_{i}(x_{1}+x_{2})(q+r)+x _{1}x_{2}(q+r)^{2}}{(px_{i}+qx_{1}+ r x_{2})(px_{i}+q x_{2}+ r x_{1})} \Biggr]. \end{aligned}$$

Hence, we obtain \(\Delta_{3}\geq 0\) for \(\boldsymbol {x}\in \mathbb{R}^{n} _{++}\) (\(n\geq 3\)). Using Proposition 4 leads to the assertion that \(\operatorname {GB}^{p,q,r}(\boldsymbol {x})\) is Schur harmonic convex on \(\mathbb{R}^{n}_{++}\).

The proof of Theorem 1 is completed. □

Remark 5

As a direct consequence of Theorem 1, taking \(r=0\) in Theorem 1 together with the identity \(\operatorname {GB}^{p,q,0}(\boldsymbol {x})=\operatorname {GB}^{p,q}(\boldsymbol {x})\), we arrive at the assertion of Corollary 1.

4 Applications

As an application of Theorem 1, we establish the following interesting inequalities for generalized geometric Bonferroni mean.

Theorem 2

Let p, q, r be non-negative real numbers with \(p+q+r \neq 0\). Then, for arbitrary \(\boldsymbol {x}\in \mathbb{R}^{n}_{++}\) (\(n\geq 3 \)),

$$ \operatorname {GB}^{p,q,r}(\boldsymbol {x}) \leq A_{n}(\boldsymbol {x}). $$
(10)

Proof

It follows from Theorem 1 that \(\operatorname {GB}^{p,q,r}(\boldsymbol {x})\) is Schur concave on \(\mathbb{R}^{n}_{++}\).

Using Lemma 1, one has

$$ \bigl( \underbrace{ A_{n}(\boldsymbol {x}), A_{n}(\boldsymbol {x}),\ldots, A_{n}(\boldsymbol {x})}_{n} \bigr) \prec (x_{1}, x_{2},\ldots, x_{n}). $$

Thus, we deduce from Definition 1 that

$$ \operatorname {GB}^{p,q,r} \bigl( A_{n}(\boldsymbol {x}), A_{n}(\boldsymbol {x}), \ldots, A_{n}(\boldsymbol {x}) \bigr) \geq \operatorname {GB}^{p,q,r}(x_{1}, x_{2},\ldots, x_{n}), $$

which implies that

$$ A_{n}(\boldsymbol {x}) \geq \operatorname {GB}^{p,q,r}(\boldsymbol {x}). $$

Theorem 2 is proved. □

Theorem 3

Let p, q, r be non-negative real numbers with \(p+q+r \neq 0\), and let c be a constant satisfying \(0\leq c< A_{n}(\boldsymbol {x})\), \((\boldsymbol {x}-c)=(x_{1}-c,x_{2}-c,\ldots,x_{n}-c)\). Then, for arbitrary \(\boldsymbol {x}\in \mathbb{R}^{n}_{++}\) (\(n\geq 3\)),

$$ \operatorname {GB}^{p,q,r }(\boldsymbol {x}-c)\leq \biggl( 1-\frac{c}{A_{n}(\boldsymbol {x})} \biggr) \operatorname {GB}^{p,q,r}(\boldsymbol {x}). $$
(11)

Proof

By the majorization relationship given in Lemma 2,

$$ \biggl( \frac{x_{1}}{\sum_{i=1}^{n}x_{i}},\ldots,\frac{x_{n}}{\sum_{i=1} ^{n}x_{i}} \biggr) \prec \biggl( \frac{x_{1}-c}{\sum_{i=1}^{n}(x_{i}-c)},\ldots,\frac{x_{n}-c}{\sum_{i=1}^{n}(x_{i}-c)} \biggr) , $$

it follows from Theorem 1 that

$$ \operatorname {GB}^{p,q,r} \biggl( \frac{x_{1}}{\sum_{i=1}^{n}x_{i}},\ldots,\frac{x_{n}}{ \sum_{i=1}^{n}x_{i}} \biggr) \geq \operatorname {GB}^{p,q,r} \biggl( \frac{x_{1}-c}{\sum_{i=1}^{n}(x_{i}-c)},\ldots,\frac{x_{n}-c}{\sum_{i=1}^{n}(x_{i}-c)} \biggr) , $$

that is,

$$ \frac{\operatorname {GB}^{p,q,r}(x_{1},x_{2},\ldots,x_{n})}{\sum_{i=1}^{n}x_{i}} \geq \frac{\operatorname {GB}^{p,q,r}(x_{1}-c,x_{2}-c,\ldots,x_{n}-c)}{\sum_{i=1} ^{n}(x_{i}-c)}, $$

which implies that

$$ \operatorname {GB}^{p,q,r }(\boldsymbol {x}-c)\leq \biggl( 1-\frac{c}{A_{n}(\boldsymbol {x})} \biggr) \operatorname {GB}^{p,q,r}(\boldsymbol {x}). $$

This completes the proof of Theorem 3. □

5 Conclusion

This paper is a follow-up study of our recent work [24], we generalize the geometric Bonferroni mean by introducing three non-negative parameters p, q, r, under the condition of \(p+q+r\neq 0\), we prove that the generalized geometric Bonferroni mean \(\operatorname {GB}^{p,q,r}(\boldsymbol {x})\) is Schur concave, Schur geometric convex and Schur harmonic convex on \(\mathbb{R}^{n}_{++}\). As an application of the Schur convexity, we establish two inequalities for generalized geometric Bonferroni mean. In fact, there have been a large number inequalities for means which originate from the Schur convexity of functions. For details, we refer the interested reader to [27–32] and the references therein.