Necessary and sufficient conditions on the Schur convexity of a bivariate mean

1.   Preliminaries

Throughout the paper, denote $\mathbb{R} = (-\infty, \infty)$, $\mathbb{R}_0 = [0, \infty)$, and $\mathbb{R}_+ = (0, \infty)$.

We recall the following definitions.

Definition 1.1 (^[5,13]). Let $\boldsymbol{x} = (x_1, x_2, \dotsc, x_{n})$ and $\boldsymbol{y} = (y_1, y_2, \dotsc, y_{n})\in\mathbb{R}^{n}$.

1. A $n$-tuple $\boldsymbol{x}$ is said to be strictly majorized by $\boldsymbol{y}$ (in symbols $\boldsymbol{x}\prec \boldsymbol{y}$) if $\sum_{i = 1}^k x_{[i]} \le \sum_{i = 1}^k y_{[i]}$ for $k = 1, 2, \dotsc, n-1$ and $\sum_{i = 1}^n x_i = \sum_{i = 1}^n y_i$, where $x_{[1]}\ge x_{[2]}\ge\dotsm \ge x_{[n]}$ and $y_{[1]}\ge y_{[2]}\ge\dotsm \ge y_{[n]}$ are rearrangements of $\boldsymbol{x}$ and $\boldsymbol{y}$ in a descending order.

2. A set $\Omega \subseteq \mathbb{R} ^{n}$ is said to be convex if $(\alpha x_1+\beta y_1, \dotsc, \alpha x_{n}+\beta y_{n}) \in\Omega$ for all $\boldsymbol{x}$ and $\boldsymbol{y} \in\Omega$, where $\alpha$ and $\beta \in[0, 1]$ with $\alpha +\beta = 1$.

3. Let $\Omega \subseteq \mathbb{R} ^{n}$. Then a function $\varphi:\Omega \to \mathbb{R}$ is said to be Schur convex (or Schur concave respectively) on $\Omega$ if the majorization $\boldsymbol{x} \prec \boldsymbol{y}$ on $\Omega$ implies the inequality $\varphi(\boldsymbol{x})\lesseqgtr\varphi (\boldsymbol{y})$.

Definition 1.2 (^[1,18]). Let $\Omega\subseteq\mathbb{R}_+^n$.

1. A set $\Omega$ 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 $\boldsymbol{xy} = \sum_{i = 1}^nx_iy_i$ and $\frac1{\boldsymbol{x}} = \bigl(\frac1{x_1}, \frac1{x_2}, \dotsc, \frac1{x_n}\bigr)$.

2. A function $\varphi:\Omega \to \mathbb{R}$ is said to be Schur harmonically convex (or Schur harmonically concave respectively) on $\Omega$ if $\frac1{\boldsymbol{x}} \prec \frac1{\boldsymbol{y}}$ implies $\varphi({\boldsymbol{x}}) \lesseqgtr\varphi(\boldsymbol{y})$.

3. A function $\psi:\Omega\to\mathbb{R}$ is called to be Schur geometrically convex on $\Omega$ if the relation $\ln \boldsymbol{\nu} \preceq \ln \boldsymbol{\vartheta}$ on $\Omega$ implies the inequality $\psi(\boldsymbol{\nu})\le\psi(\boldsymbol{\vartheta})$.

For more information on the theory of majorization, various Schur convexity, and their applications, please refer to the papers ^{[6,7,8,11,12,14,15,16,17,19]} and closely related references therein.

In 2009, Kuang defined in [4,p. 61] a bivariate mean $K_{\omega_1, \omega_2;p}(a, b)$ of positive numbers $a, b$ with three real parameters $\omega_1, \omega_2, p$ as

where $(a, b)\in\mathbb{R}^2_+$, $\omega_1, \omega_2\in\mathbb{R}_0$ with $\omega_1+\omega_2\ne0$, $A(a, b) = \frac{a+b}{2}$, and $G(a, b) = \sqrt{ab}\,$. In particular, if $\omega_1 = 2$ and $\omega_2 = \omega\ge0$, then

is the generalized Heronian mean which was introduced in ^[10] in 2008.

2.   Backgrounds and motivations

We recall the following known conclusions.

Theorem 2.1 ([2,Theorem 1.1]). Let $p\in\mathbb{R}$ and $\omega_1, \omega_2\in\mathbb{R}_0$ with $\omega_1+\omega_2\ne 0$.

1. When $\omega_1\omega_2 \ne 0$, if $p\ge 2$ and $p\bigl(\omega_1-\frac{\omega_2}{2}\bigr)-\omega_1\ge 0$, then $K_{\omega_1, \omega_2;p}(a, b)$ is Schur convex with respect to $(a, b)\in\mathbb{R}_+^2$; if $1\le p < 2$ and $p(\omega_1-\frac{\omega_2}{2})-\omega_1\le 0$, then $K_{\omega_1, \omega_2;p}(a, b)$ is Schur concave with respect to $(a, b)\in\mathbb{R}_+^2$; if $p < 1$, then $K(\omega_1, w\omega_2, p;a, b)$ is Schur concave with respect to $(a, b)\in\mathbb{R}_+^2$.

2. When $\omega_1 = 0$ and $\omega_2 \ne 0$, the mean $K_{\omega_1, \omega_2;p}(a, b)$ is Schur concave with respect to $(a, b)\in\mathbb{R}_+^2$.

3. When $\omega_1\ne 0$ and $\omega_2 = 0$, if $p\ge 1$, then $K_{\omega_1, \omega_2;p}(a, b)$ is Schur convex with respect to $(a, b)\in\mathbb{R}_+^2$; if $p < 1$, then $K_{\omega_1, \omega_2;p}(a, b)$ is Schur concave with respect to $(a, b)\in\mathbb{R}_+^2$.

Theorem 2.2 ([2,Theorem 1.2]). Let $p\in\mathbb{R}$ and $\omega_1, \omega_2\in\mathbb{R}_0$ with $\omega_1+\omega_2\ne 0$. If $p\ge 0$, then $K_{\omega_1, \omega_2;p}(a, b)$ is Schur geometrically convex with respect to $(a, b)\in\mathbb{R}_+^2$. If $p < 0$, then $K_{\omega_1, \omega_2;p}(a, b)$ is Schur geometrically concave with respect to $(a, b)\in\mathbb{R}_+^2$.

Theorem 2.3 ([2,Theorem 1.3]). Let $p\in\mathbb{R}$ and $\omega_1, \omega_2\in\mathbb{R}_0$ with $\omega_1+\omega_2\ne 0$. If $p\ge -1$, then $K_{\omega_1, \omega_2;p}(a, b)$ is Schur harmonically convex with respect to $(a, b)\in\mathbb{R}_+^2$. If $-2 < p < -1$ and $\omega_1(p+1)+\omega_2\bigl(\frac{p}{2}+1\bigr)\ge 0$, then the mean $K_{\omega_1, \omega_2;p}(a, b)$ is Schur harmonically convex with respect to $(a, b)\in\mathbb{R}_+^2$. If $p\le-2$ and $\omega_1(\frac{p}{2}+1)+\omega_2 = 0$, then $K_{\omega_1, \omega_2;p}(a, b)$ is Schur harmonically concave with respect to $(a, b)\in\mathbb{R}_+^2$.

In ^[10], the Schur convexity, Schur geometric convexity, and Schur harmonic convexity of the mean $H_{p, \omega}(a, b)$ are studied. In ^[3], sufficient and necessary conditions for $H_{p, \omega}(a, b)$ to be Schur convex and Schur harmonically convex are studied.

Theorem 2.4 ([3,Theorem 2]). Let $p\in\mathbb{R}$ and $\omega\ge 0$. Then $H_{p, \omega}(a, b)$ is

1. Schur convex on $\mathbb{R}_+^2$ if and only if $(p, \omega)\in E_1$,

2. Schur concave on $\mathbb{R}_+^2$ if and only if $(p, \omega)\in E_2$,

where $E_1$ and $E_2$ are given by

and

Theorem 2.5 ([3,Theorem 3]). Let $p\in\mathbb{R}$ and $\omega\ge 0$. Then $H_{p, \omega}(a, b)$ is

1. Schur harmonically convex on $\mathbb{R}_+^2$ if and only if $(p, \omega)\in F_1$,

2. Schur harmonically concave on $\mathbb{R}_+^2$ if and only if $(p, \omega)\in F_2$,

where $F_1$ and $F_2$ are given by

and

The main purpose of this paper is to discover necessary and sufficient conditions for the bivariate mean $K_{\omega_1, \omega_2;p}(a, b)$ to be Schur convex with respect to $(a, b)\in\mathbb{R}^2_+$ for $p\in\mathbb{R}$ and to be Schur harmonically convex with respect to $(\omega_1, \omega_2)\in\mathbb{R}_0$ with $\omega_1+\omega_2 > 0$. These new results strengthen those sufficient conditions stated in Theorems 2.1 and 2.3 mentioned above.

3.   Necessary and sufficient conditions

Now we concretely state our necessary and sufficient conditions and prove them.

Theorem 3.1. Let $p\in\mathbb{R}$ and $\omega_1, \omega_2\in\mathbb{R}_0$ with $\omega_1+\omega_2 > 0$. Then

1. the bivariate mean $K_{\omega_1, \omega_2;p}(a, b)$ is Schur convex with respect to $(a, b)\in\mathbb{R}_+^2$ if and only if $(\omega_1, \omega_2;p)\in S_1$,

2. the bivariate mean $K_{\omega_1, \omega_2;p}(a, b)$ is Schur concave with respect to $(a, b)\in\mathbb{R}_+^2$ if and only if $(\omega_1, \omega_2;p)\in S_2$,

where $S_1$ and $S_2$ are given by

and

Proof. We divide our proof into three cases.

1. When $p = 0$, we have that $K_{\omega_1, \omega_2;0}(a, b) = \sqrt{ab}\,$ which is Schur concave with respect to $(a, b)\in\mathbb{R}_+^2$ for $\{(\omega_1, \omega_2;p): p = 0, \omega_1, \omega_2\in\mathbb{R}_0, \omega_1+\omega_2 > 0\}$.

2. When $p\ne0$ and $\omega_1 = 0$, we have that $K_{0, \omega_2;p}(a, b) = \sqrt{ab}\,$ which is Schur concave with respect to $(a, b)\in\mathbb{R}_+^2$ for $\{(\omega_1, \omega_2;p): p\ne0, \omega_2 > \omega_1 = 0\}$.

3. When $p\ne0$ and $\omega_1 > 0$, if we let $\omega = \frac{2\omega_2}{\omega_1}$, then

By Theorem 2.4, we obtain that

(a) the mean $K_{\omega_1, \omega_2;p}(a, b)$ is Schur convex with respect to $(a, b)\in\mathbb{R}_+^2$ if and only if

(b) the mean $K_{\omega_1, \omega_2;p}(a, b)$ is Schur concave with respect to $(a, b)\in\mathbb{R}_+^2$ if and only if

The proof of Theorem 3.1 is complete.

Remark 3.1. If we let

and

then $S_{11}\subseteq S_1$ and $S_{21}\subseteq S_2$. This means that [2,Theorem 1.1] recited in Theorem 2.1 just only put forward a sufficient condition for $K_{\omega_1, \omega_2;p}(a, b)$ to be Schur convex. In other words, our Theorem 3.1 strengths [2,Theorem 1.1].

Theorem 3.2. Let $p\in\mathbb{R}$ and $\omega_1, \omega_2\in\mathbb{R}_0$ with $\omega_1+\omega_2 > 0$. Then

1. the bivariate mean $K_{\omega_1, \omega_2;p}(a, b)$ is Schur harmonically convex with respect to $(a, b)\in\mathbb{R}_+^2$ if and only if $(\omega_1, \omega_2;p)\in H_1$,

2. the bivariate mean $K_{\omega_1, \omega_2;p}(a, b)$ is Schur harmonically concave with respect to $(a, b)\in\mathbb{R}_+^2$ if and only if $(\omega_1, \omega_2;p)\in H_2$,

where $H_1$ and $H_2$ are given by

and

Proof. We divide our proof into two cases.

1. If $p = 0$ and $\omega_1, \omega_2\in\mathbb{R}_0$ with $\omega_1+\omega_2 > 0$, or if $p\ne0$ and $\omega_1 = 0 < \omega_2$, then $K_{\omega_1, \omega_2;p}(a, b) = \sqrt{ab}\,$ is Schur harmonically convex with respect to $(a, b)\in\mathbb{R}_+^2$.

2. If $p\ne0$ and $\omega_1 > 0$, then letting $\omega = \frac{2\omega_2}{\omega_1}$, using (3.3), and employing Theorem 2.5, we obtain

(a) the mean $K_{\omega_1, \omega_2;p}(a, b)$ is Schur harmonically convex with respect to $(a, b)\in\mathbb{R}_+^2$ if and only if

(b) the mean $K_{\omega_1, \omega_2;p}(a, b)$ is Schur harmonically concave with respect to $(a, b)\in\mathbb{R}_+^2$ if and only if

The proof of Theorem 3.2 is complete.

Remark 3.2. If letting

and

then $H_{11}\subseteq H_1$ and $H_{21}\subseteq H_2$. This means that [2,Theorem 1.3] recited in Theorem 2.3 just only put forward a sufficient condition for $K_{\omega_1, \omega_2;p}(a, b)$ to be Schur harmonically convex. In other words, our Theorem 3.2 strengths [2,Theorem 1.3].

4.   Applications

Now we apply Theorems 3.1 and 3.2 to construct several inequalities.

Theorem 4.1. Suppose $p\in\mathbb{R}$ and $\omega_1, \omega_2\in\mathbb{R}_0$ with $\omega_1+\omega_2 > 0$. Let $(a, b)\in\mathbb{R}_+^2$, $u(t) = tb+(1-t)a$, and $v(t) = ta+(1-t)b$ for $t\in[0, 1]$. Assume $\frac{1}{2}\le t_2\le t_1 \le 1$ or $0\le t_1\le t_2\le \frac{1}{2}$.

1. If $(\omega_1, \omega_2;p)\in S_1$, then

where $S_1$ is given by (3.1).

2. If $(\omega_1, \omega_2;p)\in S_2$, then

where $S_2$ is given by (3.2).

Proof. In ^[9], it was obtained that, if $a\le b$, then

Then, by a similar derivation as in the proof of [2,Theorem 4.1], one can verify required inequalities straightforwardly.

Remark 4.1. In Theorem 4.1, if $(\omega_1, \omega_2;p)\in S_1$ and $\omega_2 > 0$, then $\bigl(\frac{\omega_1 }{\omega_1+\omega_2}\bigr)^{1/p} > \frac{1}{2}$.

Theorem 4.2. Suppose $p\in\mathbb{R}$ and $\omega_1, \omega_2\in\mathbb{R}_0$ with $\omega_1+\omega_2 > 0$. Let $(a, b)\in\mathbb{R}_+^2$.

1. If $(\omega_1, \omega_2;p)\in H_1$, then

where $H_1$ is given by (3.4).

2. If $(\omega_1, \omega_2;p)\in H_2$, the double inequality (4.2) is reversed, where $H_2$ is given by (3.5).

Proof. Considering the relations in (4.1) and imitating the proof of [2,Theorem 4.6] arrive at required inequalities. The proof is complete.

5.   Conclusions

In this paper, we have found necessary and sufficient conditions in Theorems 3.1 and 3.2 for the bivariate mean $K_{\omega_1, \omega_2;p}(a, b)$ of positive numbers $a, b$ with three real parameters $\omega_1, \omega_2, p$ to be Schur convex or Schur harmonically convex respectively. These necessary and sufficient conditions have also been applied in Theorems 4.1 and 4.2.

In mathematics, necessary and sufficient condition is the best conclusion.

Acknowledgments

This work was partially supported by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia under Grant No. NJZY20119, No. NJZY19157, and No. NJZZ18154, by the Science Research Fund of Inner Mongolia University for Nationalities under Grant No. NMDYB19056 and No. NMDYB1777, and by the Research Topic of Education and Teaching of Inner Mongolia University for Nationalities under Grant No. YB2019015 and No. YB2019016, China.

The authors appreciate anonymous referees for their careful corrections to and valuable comments on the original version of this paper.

Authors' contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Conflict of interest

The authors declare that they have no conflict of interest.