Abstract
In this paper, using the properties of Schurconvex function, Schurgeometrically convex function and Schurharmonically convex function, we provide much simpler proofs of the Schurconvexity, Schurgeometric convexity and Schurharmonic convexity for a composite function of the complete symmetric function.
Similar content being viewed by others
Background
Throughout the article, \(\ {\mathbb {R}}\) denotes the set of real numbers, \(\varvec{x} = (x_1, x_2, \ldots , x_n)\) denotes ntuple (ndimensional real vectors), the set of vectors can be written as
In particular, the notations \({\mathbb {R}}\) and \({\mathbb {R}}_{+}\) denote \({\mathbb {R}}^{1}\) and \({\mathbb {R}}^{1}_{+}\), respectively.
The following complete symmetric function is an important class of symmetric functions.
For \(\varvec{x}=(x_1,x_2,\ldots ,x_n) \in {\mathbb {R}}^n\), the complete symmetric function \(c_n(\varvec{x},r)\) is defined as
where \(c_0(\varvec{x},r)=1, \,r\in \{1,2,\ldots , n \},\,\) \(i_1,i_2,\ldots , i_n\) are nonnegative integers.
It has been investigated by many mathematicians and there are many interesting results in the literature.
Guan (2006) discussed the Schurconvexity of \(c_n(\varvec{x},r)\) and proved that \(c_n(\varvec{x},r)\) is increasing and Schurconvex on \({\mathbb {R}}^n_+\). Subsequently, Chu et al. (2011) proved that \(c_n(\varvec{x},r)\) is Schurgeometrically convex and harmonically convex on \({\mathbb {R}}^n_+\).
Recently, Sun et al. (2014) studied the Schurconvexity, Schurgeometric convexity and Schurharmonic convexity of the following composite function of \(c_n(\varvec{x},r)\)
Using the Lemma 1, Lemma 2 and Lemma 3 in second section, they proved as follows: Theorem A, Theorem B and Theorem C, respectively.
Theorem A
For \(\varvec{x}=(x_1,x_2,\ldots ,x_n) \in [0,1)^n\cup (1,+\infty )^n\) and \(r \in {\mathbb {N}}\) ,

(i) \(F_n(\varvec{x},r)\) is increasing in \(x_{i}\) for all i \(\in\) \(\{1, 2, \ldots , n\}\) and Schurconvex on \([0, 1)^n\) for each r fixed;

(ii) if r is even integer (or odd integer, respectively), then \(F_n(\varvec{x},r)\) is Schurconvex (or Schurconcave, respectively) on \((1,+\infty )^n\) , and it is decreasing (or increasing, respectively) in \(x_{i}\) for all \(i \in \{1,2,\ldots ,n\}\).
Theorem B
For \(\varvec{x}=(x_1,x_2,\ldots ,x_n) \in [0,1)^n\cup (1,+\infty )^n\) and \(r \in \mathbb {N}\) ,

(i) \(F_n(\varvec{x},r)\) is Schurgeometrically convex on \([0, 1)^n\) ;

(ii) if r is even integer (or odd integer, respectively), then \(F_n(\varvec{x},r)\) is Schurgeometrically convex (or Schurgeometrically concave, respectively) on \((1,+\infty )^n\).
Theorem C
For \(\varvec{x}=(x_1,x_2,\ldots ,x_n) \in [0,1)^n\cup (1,+\infty )^n\) and \(r \in \mathbb {N}\) ,

(i) \(F_n(\varvec{x},r)\) is Schurharmonically convex on \([0, 1)^n\) ;

(ii) if r is even integer (or odd integer, respectively), then \(F_n(\varvec{x},r)\) is Schurharmonically convex (or Schurharmonically concave, respectively) on \((1,+\infty )^n\).
In this paper, using the properties of Schurconvex function, Schurgeometrically convex function and Schurharmonically convex function, we will provide much simpler proofs of the above results.
Definitions and lemmas
For convenience, we recall some definitions as follows.
Definition 1
Let \(\varvec{x} = ( x_{1},x_{2},\ldots , x_{n })\) and \(\varvec{y} = ( y_{1},y_{2},\ldots , y_{n }) \in {\mathbb {R}}^{n}\).

(i) \(\varvec{x}\ge \varvec{y}\) means \(x_{i} \ge y_{i}\) for all \(i=1, 2, \ldots , n\).

(ii) Let \(\Omega \subset {\mathbb {R}} ^{n}\), \(\varphi\): \(\Omega \rightarrow {\mathbb {R}}\) is said to be increasing if \(\varvec{x} \ge \varvec{y}\) implies \(\varphi {(\varvec{x})} \ge \varphi {(\varvec{y})}\). \(\varphi\) is said to be decreasing if and only if \(\varphi\) is increasing.
Definition 2
Let \(\varvec{x} = ( x_{1},x_{2},\ldots , x_{n })\) and \(\varvec{y} = ( y_{1},y_{2},\ldots , y_{n }) \in {\mathbb {R}}^{n}\).
 (i):

\(\varvec{x}\) is said to be majorized by \(\varvec{y}\) (in symbols \(\varvec{x} \prec \varvec{y}\)) 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 \(\varvec{x}\) and \(\varvec{y}\) in a descending order.
 (ii):

Let \(\Omega \subset {\mathbb {R}}^{n}\), \(\varphi\): \(\Omega \rightarrow {\mathbb {R}}\) is said to be a Schurconvex function on \(\Omega\) if \(\varvec{x} \prec \varvec{y}\) on \(\Omega\) implies \(\varphi \left( \varvec{x} \right) \le\) \(\varphi \left( \varvec{y} \right) .\) The function \(\varphi\) is said to be Schurconcave on \(\Omega\) if and only if \( \varphi\) is a Schurconvex function on \(\Omega\).
Definition 3
Let \(\varvec{x} = ( x_{1},x_{2},\ldots , x_{n })\) and \(\varvec{y} = ( y_{1},y_{2},\ldots , y_{n }) \in {\mathbb {R}}^{n}\).

(i) \(\Omega \subset {\mathbb {R}}^{n}\) is said to be a convex set if \(\varvec{x},\varvec{y}\in \Omega , 0 \le \alpha \le 1\), implies \(\alpha \varvec{x}+(1\alpha )\varvec{y}=\left( \alpha x_1+(1\alpha )y_1,\alpha x_2+(1\alpha )y_2,\ldots ,\alpha x_n+(1\alpha )y_n\right) \in \Omega\).

(ii) Let \(\Omega \subset {\mathbb {R}}^{n}\) be a convex set. A function \(\varphi\): \(\Omega \rightarrow {\mathbb {R}}\) is said to be convex on \(\Omega\) if
$$\begin{aligned} \varphi \left( \alpha \varvec{x}+(1\alpha )\varvec{y}\right) \le \alpha \varphi (\varvec{x})+(1\alpha )\varphi (\varvec{y}) \end{aligned}$$for all \(\varvec{x},\varvec{y}\in \Omega\), and all \(\alpha \in [0,1]\). The function \(\varphi\) is said to be concave on \(\Omega\) if and only if \( \varphi\) is a convex function on \(\Omega\).
Definition 4

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

(ii) A function \(\varphi : \Omega \rightarrow {\mathbb {R}}\) is called symmetric if for every permutation matrix P, \(\varphi (\varvec{x}P) = \varphi (\varvec{x})\) for all \(\varvec{x} \in \Omega\).
Lemma 1
(Schurconvex function decision theorem) (Marshall et al. 2011, p. 84) Let \(\Omega \subset {\mathbb {R}} ^n\) be symmetric convex set with nonempty interior. \(\Omega ^0\) is the interior of \(\Omega\) . The function \(\varphi :\Omega \rightarrow {\mathbb {R}}\) is continuous on \(\Omega\) and continuously differentiable on \(\Omega ^0\) . Then \(\varphi\) is a \(Schurconvex\,(or\,Schurconcave,\,respectively)\,function\) if and only if \(\varphi\) is symmetric on \(\Omega\) and
holds for any \(\varvec{x} \in \Omega ^0\).
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 “Schurconvex”. It has many important applications in analytic inequalities, combinatorial optimization, quantum physics, information theory, and other related fields. See Marshall et al. (2011), Rovenţa (2010), Čuljak et al. (2011), Zhang and Shi (2014).
Definition 5
Let \(\Omega \subset {\mathbb {R}}_{+}^{n}\), \(\varvec{x} = ( x_{1},x_{2},\ldots , x_{n })\) and \(\varvec{y} = ( y_{1},y_{2},\ldots , y_{n }) \in {\mathbb {R}}_{+}^{n}\).
 (i):

(Zhang 2004, p. 64) \(\Omega\) 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 \(\varvec{x}\), \(\varvec{y} \in \Omega\) and \(\alpha\), \(\beta \in [0, 1]\) such that \(\alpha +\beta =1\).
 (ii):

(Zhang 2004, p. 107) The function \(\varphi\): \(\Omega \rightarrow {\mathbb {R}}_+\) is said to be a Schurgeometrically convex function on \(\Omega\), for any \(\varvec{x}, \varvec{y} \in \Omega\), if
$$\begin{aligned} (\log x_{1},\log x_{2},\ldots ,\log x_{n}) \prec (\log y_{1},\log y_{2},\ldots , \log y_{n}) \end{aligned}$$implies \(\varphi \left( \varvec{x} \right) \le \varphi \left( \varvec{y} \right)\). The function \(\varphi\) is said to be a Schurgeometrically concave function on \(\Omega\) if and only if \( \varphi\) is a Schurgeometrically convex function on \(\Omega\).
By Definition 5, the following is obvious.
Proposition 1
Let \(\Omega \subset {\mathbb {R}}_{+}^n\) , and let
Then \(\varphi :\Omega \rightarrow {\mathbb {R}}_+\) is a Schurgeometrically convex (or Schurgeometrically concave, respectively) function on \(\Omega\) if and only if \(\varphi (e^{x_1},e^{x_2},\ldots ,e^{x_n})\) is a Schurconvex (or Schurconcave, respectively) function on \(\log \Omega\).
Lemma 2
(Schurgeometrically convex function decision theorem) (Zhang 2004, p.108) Let \(\Omega \subset {\mathbb {R}}_ {+} ^n\) be a symmetric and geometrically convex set with a nonempty interior \(\Omega ^0\) . Let \(\varphi :\Omega \rightarrow {\mathbb {R}}_+\) be continuous on \(\Omega\) and differentiable in \(\Omega ^0\) . If \(\varphi\) is symmetric on \(\Omega\) and
holds for any \(\varvec{x} = \left( {x_1, x_2, \ldots ,x_n } \right) \in \Omega ^0\) , then \(\varphi\) is a Schurgeometrically convex (or Schurgeometrically concave, respectively) function.
The Schurgeometric convexity was proposed by Zhang (2004), and was investigated by Chu et al. (2008), Guan (2007), Sun et al. (2009), and so on. We also note that some authors use the term “Schur multiplicative convexity”.
In 2009, Chu (Chu et al. (2011), Chu and Sun (2010), Chu and Lv (2009)) introduced the notion of Schurharmonically convex function.
Definition 6
Chu and Sun (2010) Let \(\Omega \subset {\mathbb {R}}_{+}^{n}\), \(\varvec{x} = ( x_{1},x_{2},\ldots , x_{n })\) and \(\varvec{y} = ( y_{1},y_{2},\ldots , y_{n }) \in {\mathbb {R}}_{+}^{n}\).

(i) A set \(\Omega\) is said to be harmonically convex if \(( \frac{2 x_1 y_1}{x_1+y_1}, \frac{2 x_2 y_2}{x_2+y_2}, \ldots , \frac{2 x_n y_n}{x_n+y_n}) \in \Omega\) for every \({\varvec{x},\varvec{y}}\in \Omega\).

(ii) A function \(\varphi :\Omega \rightarrow {\mathbb {R}}_+\) is said to be Schurharmonically convex on \(\Omega\), for any \(\varvec{x}, \varvec{y} \in \Omega\), if \(( \frac{1}{x_{1}}, \frac{1}{x_{2}}, \ldots , \frac{1}{x_{n}}) \prec ( \frac{1}{y_{1}}, \frac{1}{y_{2}}, \ldots , \frac{1}{y_{n}})\) implies \(\varphi ({\varvec{x}}) \le \varphi ({\varvec{y}})\). A function \(\varphi\) is said to be a Schurharmonically concave function on \(\Omega\) if and only if \(\varphi\) is a Schurharmonically convex function on \(\Omega\).
By Definition6, the following is obvious.
Proposition 2
Let \(\Omega \subset {\mathbb {R}}_{+}^n\) be a set, and let \(\frac{1}{\Omega } = \{( \frac{1}{x_1}, \frac{1}{x_2}, \ldots , \frac{1}{x_n}) :( x_{1},x_{2},\ldots , x_{n }) \in \Omega \}\) . Then \(\varphi :\Omega \rightarrow {\mathbb {R}}_+\) is a Schurharmonically convex (or Schurharmonically concave, respectively) function on \(\Omega\) if and only if \(\varphi \left ( \frac{1}{x_1}, \frac{1}{x_2},\ldots ,\frac{1}{x_n}\right)\) is a Schurconvex (or Schurconcave, respectively) function on \(\frac{1}{\Omega }\).
Lemma 3
(Schurharmonically convex function decision theorem) (Chu and Sun 2010) Let \(\Omega \subset {\mathbb {R}}_+^n\) be a symmetric and harmonically convex set with inner points and let \(\varphi :\Omega \rightarrow {\mathbb {R}}_+\) be a continuous symmetric function which is differentiable on \(\Omega ^0\) . Then \(\varphi\) is Schurharmonically convex (or Schurharmonically concave, respectively) on \(\Omega\) if and only if
Lemma 4
If r is even integer (or odd integer, respectively), then \(c_n(\varvec{x},r)\) is decreasing and Schurconvex (or increasing and Schurconcave, respectively) on \({\mathbb {R}}^{n}_{}\) .
Proof
Notice that
i.e.
If r is even integer, then \(c_n(\varvec{x},r)=c_n(\varvec{x},r)\). For \(\varvec{x},\varvec{y}\in {\mathbb {R}}^{n}_{}\), if \(\varvec{x}\prec \varvec{y}\), then \(\varvec{x}\prec \varvec{y}\) and \(\varvec{x},\varvec{y} \in {\mathbb {R}}^{n}_{+}\), but \(c_n(\varvec{x},r)\) is Schurconvex in \({\mathbb {R}}^n_+\), so that \(c_n(\varvec{x},r)\le c_n(\varvec{y},r)\), i.e. \(c_n(\varvec{x},r)\le c_n(\varvec{y},r)\), this shows that \(c_n(\varvec{x},r)\) is Schurconvex in \({\mathbb {R}}^{n}_{}\). If \(\varvec{x}\le \varvec{y}\), then \(\varvec{x}\ge \varvec{y}\), but \(c_n(\varvec{x},r)\) is increasing in \({\mathbb {R}}^n_+\), so that \(c_n(\varvec{x},r)\ge c_n(\varvec{y},r)\), i.e. \(c_n(\varvec{x},r)\ge c_n(\varvec{y},r)\), this shows that \(c_n(\varvec{x},r)\) is decreasing in \({\mathbb {R}}^{n}_{}\).
If r is odd integer, then \(c_n(\varvec{x},r)=c_n(\varvec{x},r)\). For \(\varvec{x},\varvec{y}\in {\mathbb {R}}^{n}_{}\), if \(\varvec{x}\prec \varvec{y}\), then \(\varvec{x}\prec \varvec{y}\) and \(\varvec{x},\varvec{y} \in {\mathbb {R}}^{n}_{+}\), but \(c_n(\varvec{x},r)\) is Schurconvex in \({\mathbb {R}}^n_+\), so that \(c_n(\varvec{x},r)\le c_n(\varvec{y},r)\), i.e. \(c_n(\varvec{x},r)\ge c_n(\varvec{y},r)\), this shows that \(c_n(\varvec{x},r)\) is Schurconcave in \({\mathbb {R}}^{n}_{}\). If \(\varvec{x}\le \varvec{y}\), then \(\varvec{x}\ge \varvec{y}\), but \(c_n(\varvec{x},r)\) is increasing in \({\mathbb {R}}^n_+\), so that \(c_n(\varvec{x},r)\ge c_n(\varvec{y},r)\), i.e. \(c_n(\varvec{x},r)\le c_n(\varvec{y},r)\), this shows that \(c_n(\varvec{x},r)\) is increasing in \({\mathbb {R}}^{n}_{}\). \(\square\)
Lemma 5
(Marshall et al. 2011, p. 91; Wang 1990, p. 64–65) Let the set \({\mathbb {A}}, {\mathbb {B}}\subset {\mathbb {R}}\) , \(\varphi :{\mathbb {B}}^n\rightarrow {\mathbb {R}}\) , \(f:{\mathbb {A}}\rightarrow {\mathbb {B}}\) and \(\psi (x_1, x_2, \ldots , x_n) = \varphi (f(x_1),f(x_2), \ldots , f(x_n)):{\mathbb {A}}^n\rightarrow {\mathbb {R}}\) .

(i) If \(\varphi\) is increasing and Schurconvex and f is increasing and convex, then \(\psi\) is increasing and Schurconvex.

(ii) If \(\varphi\) is decreasing and Schurconvex and f is increasing and concave, then \(\psi\) is decreasing and Schurconvex.

(iii) If \(\varphi\) is increasing and Schurconcave and f is increasing and concave, then \(\psi\) is increasing and Schurconcave.

(iv) If \(\varphi\) is decreasing and Schurconvex and f is decreasing and concave, then \(\psi\) is increasing and Schurconvex.

(v) If \(\varphi\) is increasing and Schurconcave and f is decreasing and concave, then \(\psi\) is decreasing and Schurconcave.
Lemma 6
Let the set \(\Omega \subset {\mathbb {R}}^n_+\) . The function \(\varphi :\Omega \rightarrow {\mathbb {R}}_+\) is differentiable.

(i) If \(\varphi\) is increasing and Schurconvex, then \(\varphi\) is Schurgeometrically convex.

(ii) If \(\varphi\) is decreasing and Schurconcave, then \(\varphi\) is Schurgeometrically concave.
Proof
We only give the proof of Lemma 6 (i) in detail. Similar argument leads to the proof of Lemma 6 (ii).
For \(\varvec{x}\in I\subset {\mathbb {R}}_+\) and \(x_1 \ne x_2\), we have
Since \(\varphi\) is Schurconvex on \(\Omega\), by Lemma 1, we have
Notice that \(\varphi\) and \(y=\log x\) is increasing, we have \(\frac{\partial \varphi }{\partial x_2}\ge 0\), \(\frac{\log x_1 \log x_2}{x_1  x_2 }\ge 0\) and \(\left( x_1  x_2 \right) \left( \log x_1 \log x_2 \right) \ge 0\), so that \(\Delta \ge 0\), by Lemma 2, it follows that \(\varphi\) is Schurgeometrically convex on \(\Omega\). \(\square\)
Lemma 7
Let the set \(\Omega \subset {\mathbb {R}}^n_+\) . The function \(\varphi :\Omega \rightarrow {\mathbb {R}}_+\) is differentiable.

(i) If \(\varphi\) is increasing and Schurconvex, then \(\varphi\) is Schurharmonically convex.

(ii) If \(\varphi\) is decreasing and Schurconcave, then \(\varphi\) is Schurharmonically concave.
Proof
We only give the proof of Lemma 7 (ii) in detail. Similar argument leads to the proof of Lemma 7 (i).
For \(\varvec{x}\in I\subset {\mathbb {R}}_+\) and \(x_1 \ne x_2\), we have
Since \(\varphi\) is Schurconcave on \(\Omega\), by Lemma 1, we have
Notice that \(\varphi\) is decreasing and \(y=x^2(x>0)\) is increasing, we have \(\frac{\partial \varphi }{\partial x_2}\le 0\) and \(\left( x_1  x_2 \right) \left( x^2_1  x^2_2 \right) \ge 0\), so that \(\Lambda \le 0\), by Lemma 3, it follows that \(\varphi\) is Schurharmonically concave on \(\Omega\).\(\square\)
Simple proof of theorems
Proof of Theorem A
Let \(g(t)= \frac{t}{1t}\). Directly calculating yields \(g'(t)= \frac{1}{(1t)^2}\) and \(g''(t)= \frac{2}{(1t)^3}\), it is to see that g is increasing and convex on (0, 1) and g is increasing and concave on \((1,+\infty )\).
Since \(c_n(\varvec{x},r)\) is increasing and Schurconvex in \({\mathbb {R}}^n_+\), from Lemma 5 (i) it follows that \(F_n(\varvec{x},r)\) is increasing and Schurconvex in \((0, 1)^n\), and then by continuity of \(F_n(\varvec{x},r)\) on \([0, 1)^n\), it follows that \(F_n(\varvec{x},r)\) is increasing and Schurconvex on \([0, 1)^n\).
If r is even integer, then from Lemma 4, we known that \(c_n(\varvec{x},r)\) is decreasing and Schurconvex, moreover g is increasing and concave on \((1,+\infty )\). By Lemma 5 (ii), it follows that \(F_n(\varvec{x},r)\) is decreasing and Schurconvex.
If r is odd integer, then from Lemma 4, we known that \(c_n(\varvec{x},r)\) is increasing and Schurconcave, moreover g is increasing and concave on \((1,+\infty )\). By Lemma 5 (iii), it follows that \(F_n(\varvec{x},r)\) is increasing and Schurconcave.
The proof of Theorem A is completed. \(\square\)
Proof of Theorem B
From Theorem A (i) and Lemma 6 (i), it follows that Theorem B (i) holds.
Considing
Let \(h(t)= \frac{e^t}{1e^t}\). Then \(h<0\) on \((0,+\infty )\). Directly calculating yields \(h'(t)= \frac{e^t}{(1e^t)^2}\) and \(h''(t)= \frac{e^{t}(1+e^t)}{(1e^t)^3}\), it is to see that h is increasing and concave on \((0,+\infty )\). From Lemma 4 and Lemma 5 (ii) (or (iii), respectively), it follows that if r is even integer (or odd integer, respectively), then \(F_n(e^{\varvec{x}},r)\) is Schurconvex (or Schurconcave, respectively) on \((0,+\infty )\). And then, by Proposition 1, Theorem B (ii) holds.
The proof of Theorem B is completed. \(\square\)
Proof of Theorem C
From Theorem A (i) and Lemma 7 (i), it follows that Theorem C (i) holds.
Considing
Let \(p(t)= \frac{1}{t1}\). Then \(p<0\) on (0, 1). Directly calculating yields \(p'(t)= \frac{1}{(t1)^2}\) and \(p''(t)= \frac{2}{(t1)^3}\), it is to see that p is decreasing and concave on (0, 1). From Lemma 4 and Lemma 5 (iv) (or (v), respectively), it follows that if r is even integer (or odd integer, respectively), then \(F_n\left( \frac{1}{\varvec{x}},r\right)\) is Schurconvex (or Schurconcave, respectively) on (0, 1). And then, by Proposition 2, Theorem C (ii) holds.
The proof of Theorem C is completed. \(\square\)
References
Chu YM, Lv YP (2009) The Schur harmonic convexity of the Hamy symmetric function and its applications. J Inequal Appl 2009:838529. doi:10.1155/2009/838529
Chu YM, Sun TC (2010) The Schur harmonic convexity for a class of symmetric functions. Acta Math Sci 30B(5):1501–1506
Chu YM, Zhang XM, Wang GD (2008) The Schur geometrical convexity of the extended mean values. J Convex Anal 15(4):707–718
Chu YM, Wang GD, Zhang XH (2011) The Schur multiplicative and harmonic convexities of the complete symmetric function. Math Nachr 284(5–6):653–663
Čuljak V, Franjić I, Ghulam R, Pečarić J (2011) Schurconvexity of averages of convex functions. J Inequal Appl 2011:581918. doi:10.1155/2011/581918
Guan KZ (2006) Schurconvexity of the complete symmetric function. Math Inequal Appl 9(4):567–576
Guan KZ (2007) A class of symmetric functions for multiplicatively convex function. Math Inequal Appl 10(4):745–753
Marshall AW, Olkin I, Arnold BC (2011) Inequalities: theory of majorization and its application, 2nd edn. Springer, New York
Rovenţa I (2010) Schur convexity of a class of symmetric functions. Ann Univ Craiova Math Comput Sci Ser 37(1):12–18
Sun MB, Chen NB, Li SH (2014) Some properties of a class of symmetric functions and its applications. Math Nachr 287(13):1530–1544. doi:10.1002/mana.201300073
Sun TC, Lv YP, Chu YM (2009) Schur multiplicative and harmonic convexities of generalized Heronian mean in \(n\) variables and their applications. Int J Pure Appl Math 55(1):25–33
Wang BY (1990) Foundations of majorization inequalities. Beijing Normal University Press, Beijing (in Chinese)
Zhang XM (2004) Geometrically convex functions. An’hui University Press, Hefei (in Chinese)
Zhang J, Shi HN (2014) Two double inequalities for \(k\)gamma and \(k\)Riemann zeta functions. J Inequal Appl 2014:191. doi:10.1186/1029242X2014191
Authors’ contributions
The main idea of this paper was proposed by HNS. This work was carried out in collaboration between all authors. All authors read and approved the final manuscript.
Acknowledgements
The work was supported by the Importation and Development of HighCaliber Talents Project of Beijing Municipal Institutions (Grant No. IDHT201304089) and the National Natural Science Foundation of China (Grant No. 11501030). The authors are indebted to the referees for their helpful suggestions.
Competing interests
The authors declare that they have no competing interests.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Shi, HN., Zhang, J. & Ma, QH. Schurconvexity, Schurgeometric and Schurharmonic convexity for a composite function of complete symmetric function. SpringerPlus 5, 296 (2016). https://doi.org/10.1186/s400640161940z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s400640161940z