• Research Article
• Open Access

# A new order-preserving average function on a quotient space of strictly monotone functions and its applications

Journal of Inequalities and Applications20142014:450

https://doi.org/10.1186/1029-242X-2014-450

• Received: 23 June 2014
• Accepted: 21 October 2014
• Published:

## Abstract

We introduce an order in a quotient space of strictly monotone continuous functions on a real interval and show that a new average function on this quotient space is order-preserving. We also apply this new order-preserving function to derive a finite form of Jensen type inequality with negative weights.

MSC:39B62, 26B25, 26A51.

## Keywords

• Jensen’s inequality
• strictly monotone function
• order-preserving average function

## 1 Introduction and main results

This is meant as a continuation of our paper [1] related to Jensen’s inequality [2]. The reader should refer to the recent paper of József [3] on Jensen’s inequality. Further the paper is related to the notion of quasi-arithmetic means, so the reader should refer to the recent paper of Janusz [4].

Let I be a finite closed interval $\left[m,M\right]$ on R and $C\left(I\right)$ the space of all continuous real-valued functions defined on I. Moreover, let ${C}_{\mathrm{sm}}^{+}\left(I\right)$ (resp. ${C}_{\mathrm{sm}}^{-}\left(I\right)$) be the set of all functions in $C\left(I\right)$ which are strictly monotone increasing (resp. decreasing) on I. Put
${C}_{\mathrm{sm}}\left(I\right)={C}_{\mathrm{sm}}^{+}\left(I\right)\cup {C}_{\mathrm{sm}}^{-}\left(I\right).$

Then ${C}_{\mathrm{sm}}\left(I\right)$ is equal to the space of all strictly monotone continuous functions on I. For any $\phi ,\psi \in {C}_{\mathrm{sm}}\left(I\right)$, we write $\phi \cong \psi$ if there exist two numbers $a,b\in \mathbf{R}$ such that $\phi \left(x\right)=a\psi \left(x\right)+b$ for all $x\in I$. Then it is clear that is an equivalence relation in ${C}_{\mathrm{sm}}\left(I\right)$. Let ${\stackrel{˜}{C}}_{\mathrm{sm}}\left(I\right)$ be the quotient space of ${C}_{\mathrm{sm}}\left(I\right)$ by and we denote by $\stackrel{˜}{\phi }$ the coset of $\phi \in {C}_{\mathrm{sm}}\left(I\right)$. We introduce an order in ${\stackrel{˜}{C}}_{\mathrm{sm}}\left(I\right)$ in the next section (see Theorem 2).

Let $\left(\mathrm{\Omega },\mu \right)$ be a probability space and f a function in ${L}^{1}\left(\mathrm{\Omega },\mu \right)$ such that $f\left(\omega \right)\in I$ for almost all $\omega \in \mathrm{\Omega }$. Then we see that $\phi \circ f\in {L}^{1}\left(\mathrm{\Omega },\mu \right)$ for all $\phi \in {C}_{\mathrm{sm}}\left(I\right)$ because φ is a bounded continuous function and μ is a finite measure. Put
${M}_{\phi }\left(f\right)={\phi }^{-1}\left(\int \phi \circ f\phantom{\rule{0.2em}{0ex}}d\mu \right)$

for each $\phi \in {C}_{\mathrm{sm}}\left(I\right)$. Then [[1], Theorem 1] which gives a new interpretation of Jensen’s inequality is restated as $\stackrel{˜}{\phi }⪯\stackrel{˜}{\psi }⇒{M}_{\phi }\left(f\right)\le {M}_{\psi }\left(f\right)$. In this paper, we give a new order-preserving average function ${N}_{\left[I,f\right]}$ on the quotient space ${\stackrel{˜}{C}}_{\mathrm{sm}}\left(I\right)$, according to this idea. We also apply this function ${N}_{\left[I,f\right]}$ to derive a finite form of Jensen type inequality with negative weights.

Let φ be an arbitrary function of ${C}_{\mathrm{sm}}\left(I\right)$. Since $\phi \left(I\right)$ is an interval of R and μ is a probability measure on Ω, it follows that
$\phi \left(m\right)+\phi \left(M\right)-\int \phi \circ f\phantom{\rule{0.2em}{0ex}}d\mu \in \phi \left(I\right),$
and hence we have
${\phi }^{-1}\left(\phi \left(m\right)+\phi \left(M\right)-\int \phi \circ f\phantom{\rule{0.2em}{0ex}}d\mu \right)\in I.$
Note that a simple computation implies that if $\phi ,\psi \in {C}_{\mathrm{sm}}\left(I\right)$ satisfy $\stackrel{˜}{\phi }=\stackrel{˜}{\psi }$, then
${\phi }^{-1}\left(\phi \left(m\right)+\phi \left(M\right)-\int \phi \circ f\phantom{\rule{0.2em}{0ex}}d\mu \right)={\psi }^{-1}\left(\psi \left(m\right)+\psi \left(M\right)-\int \psi \circ f\phantom{\rule{0.2em}{0ex}}d\mu \right)$

holds. Then denote by ${N}_{\left[I,f\right]}\left(\stackrel{˜}{\phi }\right)$ the above value.

In this case, our main result can be stated as follows.

Theorem 1 ${N}_{\left[I,f\right]}$ is an order-preserving real-valued function on the quotient space ${\stackrel{˜}{C}}_{\mathrm{sm}}\left(I\right)$ with order , that is, $\stackrel{˜}{\phi }⪯\stackrel{˜}{\psi }⇒{N}_{\left[I,f\right]}\left(\stackrel{˜}{\phi }\right)\le {N}_{\left[I,f\right]}\left(\stackrel{˜}{\psi }\right)$.

The above theorem easily implies the following result, which is a finite form of Jensen type inequality with negative weights.

Corollary 1 Let $\phi ,\psi \in {C}_{\mathrm{sm}}\left(I\right)$ with $\stackrel{˜}{\phi }⪯\stackrel{˜}{\psi }$ and ${t}_{1},\dots ,{t}_{n}\in \mathbf{R}$ with ${\sum }_{i=1}^{n}{t}_{i}=1$, $0<{t}_{1}$, ${t}_{n}<1$, and ${t}_{2},\dots ,{t}_{n-1}<0$. Then
${\phi }^{-1}\left(\sum _{i=1}^{n}{t}_{i}\phi \left({x}_{i}\right)\right)\le {\psi }^{-1}\left(\sum _{i=1}^{n}{t}_{i}\psi \left({x}_{i}\right)\right)$

holds for all ${x}_{1},\dots ,{x}_{n}\in I$ with ${x}_{1}\le {x}_{2},\dots ,{x}_{n-1}\le {x}_{n}$.

Finally, we give concrete examples of Corollary 1.

## 2 An order in the quotient space ${\stackrel{˜}{C}}_{\mathrm{sm}}\left(I\right)$

Let us start with the following two lemmas.

Lemma 1 Let $\phi \in {C}_{\mathrm{sm}}\left(I\right)$. Then:
1. (i)

φ is increasing and convex on I if and only if ${\phi }^{-1}$ is increasing and concave on $\phi \left(I\right)$.

2. (ii)

φ is increasing and concave on I if and only if ${\phi }^{-1}$ is increasing and convex on $\phi \left(I\right)$.

3. (iii)

φ is decreasing and convex on I if and only if ${\phi }^{-1}$ is decreasing and convex on $\phi \left(I\right)$.

4. (iv)

φ is decreasing and concave on I if and only if ${\phi }^{-1}$ is decreasing and concave on $\phi \left(I\right)$.

Proof Straightforward. □

Lemma 2
1. (i)

If φ is a convex function on I and ψ is an increasing convex function on $\phi \left(I\right)$, then $\psi \circ \phi$ is convex on I.

2. (ii)

If φ is a convex function on I and ψ is a decreasing concave function on $\phi \left(I\right)$, then $\psi \circ \phi$ is concave on I.

3. (iii)

If φ is a concave function on I and ψ is an increasing concave function on $\phi \left(I\right)$, then $\psi \circ \phi$ is concave on I.

4. (iv)

If φ is a concave function on I and ψ is a decreasing convex function on $\phi \left(I\right)$, then $\psi \circ \phi$ is convex on I.

Proof Straightforward. □

For any $\phi ,\psi \in {C}_{\mathrm{sm}}\left(I\right)$, we write $\phi ⪯\psi$ if any of the following four conditions holds:
1. (i)

$\phi ,\psi \in {C}_{\mathrm{sm}}^{+}\left(I\right)$ and $\phi \circ {\psi }^{-1}$ is concave on $\psi \left(I\right)$.

2. (ii)

$\phi \in {C}_{\mathrm{sm}}^{-}\left(I\right)$, $\psi \in {C}_{\mathrm{sm}}^{+}\left(I\right)$ and $\phi \circ {\psi }^{-1}$ is convex on $\psi \left(I\right)$.

3. (iii)

$\phi ,\psi \in {C}_{\mathrm{sm}}^{-}\left(I\right)$ and $\phi \circ {\psi }^{-1}$ is convex on $\psi \left(I\right)$.

4. (iv)

$\phi \in {C}_{\mathrm{sm}}^{+}\left(I\right)$, $\psi \in {C}_{\mathrm{sm}}^{-}\left(I\right)$ and $\phi \circ {\psi }^{-1}$ is concave on $\psi \left(I\right)$.

Remark Lemma 1 guarantees that the above $\phi ⪯\psi$ is a restatement of the concepts appearing in [[1], Lemma 3].

Lemma 3 Let $\phi ,{\phi }^{\prime },\psi ,{\psi }^{\prime }\in {C}_{\mathrm{sm}}\left(I\right)$. If $\phi \cong {\phi }^{\prime }$, $\psi \cong {\psi }^{\prime }$, and $\phi ⪯\psi$, then ${\phi }^{\prime }⪯{\psi }^{\prime }$.

Proof Assume that $\phi \cong {\phi }^{\prime }$, $\psi \cong {\psi }^{\prime }$, and $\phi ⪯\psi$. Then we must show ${\phi }^{\prime }⪯{\psi }^{\prime }$. Since $\phi \cong {\phi }^{\prime }$, $\psi \cong {\psi }^{\prime }$, we can write ${\phi }^{\prime }$ and ${\psi }^{\prime }$ as follows:
${\phi }^{\prime }=a\phi +b\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\psi }^{\prime }=c\psi +d$
for some $a,b,c,d\in \mathbf{R}$. Then we have $a\ne 0$ and $c\ne 0$. Put
$\zeta \left(x\right)=ax+b\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\eta \left(x\right)=cx+d$
for each $x\in \mathbf{R}$. In the case of $\phi ,\psi \in {C}_{\mathrm{sm}}^{+}\left(I\right)$ and $a,c>0$, we find that $\phi \circ {\psi }^{-1}$ is concave on $\psi \left(I\right)$ because $\phi ⪯\psi$. Then $\zeta \circ \phi \circ {\psi }^{-1}$ is increasing and concave on $\psi \left(I\right)$ from Lemma 2-(iii) and hence ${\phi }^{\prime }\circ {\psi }^{\prime -1}=\zeta \circ \phi \circ {\psi }^{-1}\circ {\eta }^{-1}$ is also concave on ${\psi }^{\prime }\left(I\right)$ from Lemma 2-(iii). However, since ${\phi }^{\prime },{\psi }^{\prime }\in {C}_{\mathrm{sm}}^{+}\left(I\right)$, we obtain ${\phi }^{\prime }⪯{\psi }^{\prime }$ as required. Moreover, we can easily see that ${\phi }^{\prime }⪯{\psi }^{\prime }$ holds in the other 15 cases:
$\begin{array}{c}\left[\phi \in {C}_{\mathrm{sm}}^{+}\left(I\right),\psi \in {C}_{\mathrm{sm}}^{-}\left(I\right),a>0,c>0\right],\phantom{\rule{2em}{0ex}}\dots ,\hfill \\ \left[\phi \in {C}_{\mathrm{sm}}^{-}\left(I\right),\psi \in {C}_{\mathrm{sm}}^{-}\left(I\right),a<0,c<0\right].\hfill \end{array}$

□

For any $\stackrel{˜}{\phi },\stackrel{˜}{\psi }\in {\stackrel{˜}{C}}_{\mathrm{sm}}\left(I\right)$, we write $\stackrel{˜}{\phi }⪯\stackrel{˜}{\psi }$ by the same notation if $\phi ⪯\psi$ holds. This is well defined by Lemma 3. In this case, we have the following.

Theorem 2 is an order relation in ${\stackrel{˜}{C}}_{\mathrm{sm}}\left(I\right)$.

Proof We show the theorem by dividing into three steps.
1. (I)

It is evident that satisfies the reflexivity.

2. (II)
Assume that $\stackrel{˜}{\phi }⪯\stackrel{˜}{\psi }$ and $\stackrel{˜}{\psi }⪯\stackrel{˜}{\phi }$. Then $\phi ⪯\psi$ and $\psi ⪯\phi$ hold. In the case of $\phi ,\psi \in {C}_{\mathrm{sm}}^{+}\left(I\right)$, we find that $\phi \circ {\psi }^{-1}$ is concave on $\psi \left(I\right)$ and $\psi \circ {\phi }^{-1}$ is concave on $\phi \left(I\right)$. Since $\psi \circ {\phi }^{-1}$ is increasing and concave on $\phi \left(I\right)$, it follows from Lemma 1-(ii) that $\phi \circ {\psi }^{-1}={\left(\psi \circ {\phi }^{-1}\right)}^{-1}$ is convex on $\psi \left(I\right)$. Therefore $\phi \circ {\psi }^{-1}$ is affine on $\psi \left(I\right)$ and hence $\phi \cong \psi$, that is, $\stackrel{˜}{\phi }=\stackrel{˜}{\psi }$. By the same method, we can easily see that $\stackrel{˜}{\phi }=\stackrel{˜}{\psi }$ holds in the other three cases:
$\left[\phi \in {C}_{\mathrm{sm}}^{+}\left(I\right),\psi \in {C}_{\mathrm{sm}}^{-}\left(I\right)\right],\phantom{\rule{2em}{0ex}}\left[\phi \in {C}_{\mathrm{sm}}^{-}\left(I\right),\psi \in {C}_{\mathrm{sm}}^{+}\left(I\right)\right]\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\left[\phi ,\psi \in {C}_{\mathrm{sm}}^{-}\left(I\right)\right].$

Therefore satisfies the symmetry law.
1. (III)
Assume that $\stackrel{˜}{\phi }⪯\stackrel{˜}{\psi }$ and $\stackrel{˜}{\psi }⪯\stackrel{˜}{\lambda }$. Then $\phi ⪯\psi$ and $\psi ⪯\lambda$ hold. In the case of $\phi ,\psi ,\lambda \in {C}_{\mathrm{sm}}^{+}\left(I\right)$, we find that $\phi \circ {\psi }^{-1}$ is increasing and concave on $\psi \left(I\right)$ and $\psi \circ {\lambda }^{-1}$ is concave on $\lambda \left(I\right)$. Then it follows from Lemma 2-(iii) that $\phi \circ {\lambda }^{-1}=\left(\phi \circ {\psi }^{-1}\right)\circ \left(\psi \circ {\lambda }^{-1}\right)$ is concave on $\lambda \left(I\right)$, and hence $\phi ⪯\lambda$, that is, $\stackrel{˜}{\phi }⪯\stackrel{˜}{\lambda }$ holds. By the same method, we can easily see that $\stackrel{˜}{\phi }⪯\stackrel{˜}{\lambda }$ holds in the other seven cases:
$\begin{array}{c}\left[\phi \in {C}_{\mathrm{sm}}^{+}\left(I\right),\psi \in {C}_{\mathrm{sm}}^{+}\left(I\right),\lambda \in {C}_{\mathrm{sm}}^{-}\left(I\right)\right],\phantom{\rule{2em}{0ex}}\dots ,\hfill \\ \left[\phi \in {C}_{\mathrm{sm}}^{-}\left(I\right),\psi \in {C}_{\mathrm{sm}}^{-}\left(I\right),\lambda \in {C}_{\mathrm{sm}}^{-}\left(I\right)\right].\hfill \end{array}$

Therefore satisfies the transitive law. □

## 3 Proofs of Theorem 1 and Corollary 1

Let φ be an arbitrary function of ${C}_{\mathrm{sm}}\left(I\right)$. Then an easy observation implies that
${\left(-\phi \right)}^{-1}\left(y\right)={\phi }^{-1}\left(-y\right)$
(1)
for all $y\in -\phi \left(I\right)$ and that
${N}_{\left[I,f\right]}\left(\stackrel{˜}{-\phi }\right)={N}_{\left[I,f\right]}\left(\stackrel{˜}{\phi }\right).$
(2)
Lemma 4 Let $\phi \in {C}_{\mathrm{sm}}\left(I\right)$. If either φ is increasing and concave on I or decreasing and convex on I, then
${N}_{\left[I,f\right]}\left(\stackrel{˜}{\phi }\right)\le \int {\phi }^{-1}\circ \left(\phi \left(m\right)+\phi \left(M\right)-\phi \circ f\right)\phantom{\rule{0.2em}{0ex}}d\mu \le m+M-\int f\phantom{\rule{0.2em}{0ex}}d\mu$

holds. If either φ is increasing and convex on I or decreasing and concave on I, then the above inequalities are reversed.

Proof (I) Suppose that φ is increasing and concave on I. Then ${\phi }^{-1}$ is increasing and convex on $\phi \left(I\right)$ by Lemma 1-(ii), and hence the first inequality in Lemma 4 follows from Jensen’s inequality. Put
${\phi }^{\mathrm{♯}}\left(x\right)={\phi }^{-1}\left(\phi \left(m\right)+\phi \left(M\right)-\phi \left(x\right)\right)+x$
for each $x\in I$. Then it follows from Lemma 2-(i) that ${\phi }^{\mathrm{♯}}$ is a convex function on I such that ${\phi }^{\mathrm{♯}}\left(m\right)={\phi }^{\mathrm{♯}}\left(M\right)=m+M$. Therefore we have
${\phi }^{-1}\left(\phi \left(m\right)+\phi \left(M\right)-\phi \left(f\left(\omega \right)\right)\right)\le m+M-f\left(\omega \right)$
(3)

for almost all $\omega \in \mathrm{\Omega }$. By integrating (3) with respect to ω, we obtain the second inequality in Lemma 4. We next suppose that φ is decreasing and convex on I. Then −φ is increasing and concave on I. Therefore the desired inequality follows from (1), (2), and the above argument.

(II) Suppose that φ is increasing and convex on I. Then ${\phi }^{-1}$ is increasing and concave on $\phi \left(I\right)$ by Lemma 1-(i), and hence the first inequality in Lemma 4 is reversed from Jensen’s inequality. Also since ${\phi }^{\mathrm{♯}}$ is concave on I by Lemma 2-(iii), it follows that the second inequality in Lemma 4 is reversed from a consideration in (I). Similarly for the decreasing and concave case. □

Proof of Theorem 1 Let $\stackrel{˜}{\phi },\stackrel{˜}{\psi }\in {\stackrel{˜}{C}}_{\mathrm{sm}}\left(I\right)$ with $\stackrel{˜}{\phi }⪯\stackrel{˜}{\psi }$, where $\phi ,\psi \in {C}_{\mathrm{sm}}\left(I\right)$.

(I-i) In the case of $\phi ,\psi \in {C}_{\mathrm{sm}}^{+}\left(I\right)$, we find that $\phi \circ {\psi }^{-1}$ is increasing and concave on $\psi \left(I\right)=\left[\psi \left(m\right),\psi \left(M\right)\right]$ because $\phi ⪯\psi$. Therefore we have from Lemma 4
$\begin{array}{c}\psi \left({N}_{\left[I,f\right]}\left(\stackrel{˜}{\phi }\right)\right)\hfill \\ \phantom{\rule{1em}{0ex}}=\left(\psi \circ {\phi }^{-1}\right)\left(\phi \left(m\right)+\phi \left(M\right)-\int \phi \circ f\phantom{\rule{0.2em}{0ex}}d\mu \right)\hfill \\ \phantom{\rule{1em}{0ex}}={\left(\phi \circ {\psi }^{-1}\right)}^{-1}\left(\left(\phi \circ {\psi }^{-1}\right)\left(\psi \left(m\right)\right)+\left(\phi \circ {\psi }^{-1}\right)\left(\psi \left(M\right)\right)-\int \left(\phi \circ {\psi }^{-1}\right)\circ \left(\psi \circ f\right)\phantom{\rule{0.2em}{0ex}}d\mu \right)\hfill \\ \phantom{\rule{1em}{0ex}}={N}_{\left[\psi \left(I\right),\psi \circ f\right]}\left(\stackrel{˜}{\phi \circ {\psi }^{-1}}\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le \psi \left(m\right)+\psi \left(M\right)-\int \psi \circ f\phantom{\rule{0.2em}{0ex}}d\mu \hfill \\ \phantom{\rule{1em}{0ex}}=\psi \left({N}_{\left[I,f\right]}\left(\stackrel{˜}{\psi }\right)\right),\hfill \end{array}$

so we obtain ${N}_{\left[I,f\right]}\left(\stackrel{˜}{\phi }\right)\le {N}_{\left[I,f\right]}\left(\stackrel{˜}{\psi }\right)$ since ψ is strictly increasing on I.

(I-ii) In the case of $\phi \in {C}_{\mathrm{sm}}^{-}\left(I\right)$ and $\psi \in {C}_{\mathrm{sm}}^{+}\left(I\right)$, we find that $\phi \circ {\psi }^{-1}$ is decreasing and convex on $\psi \left(I\right)$ because $\phi ⪯\psi$. Then $-\phi ,\psi \in {C}_{\mathrm{sm}}^{+}\left(I\right)$ and $\left(-\phi \right)\circ {\psi }^{-1}$ is increasing and concave on $\psi \left(I\right)$. Therefore we have from (I-i) and (2)
${N}_{\left[I,f\right]}\left(\stackrel{˜}{\phi }\right)={N}_{\left[I,f\right]}\left(\stackrel{˜}{-\phi }\right)\le {N}_{\left[I,f\right]}\left(\stackrel{˜}{\psi }\right).$
(I-iii) In the case of $\phi ,\psi \in {C}_{\mathrm{sm}}^{-}\left(I\right)$, we find that $\phi \circ {\psi }^{-1}$ is increasing and convex on $\psi \left(I\right)$ because $\phi ⪯\psi$. Then $\phi \in {C}_{\mathrm{sm}}^{-}\left(I\right)$, $-\psi \in {C}_{\mathrm{sm}}^{+}\left(I\right)$, and $\phi \circ {\left(-\psi \right)}^{-1}$ is decreasing and convex on $-\psi \left(I\right)$ by (1). Therefore we have from (I-ii) and (2)
${N}_{\left[I,f\right]}\left(\stackrel{˜}{\phi }\right)\le {N}_{\left[I,f\right]}\left(\stackrel{˜}{-\psi }\right)={N}_{\left[I,f\right]}\left(\stackrel{˜}{\psi }\right).$
(I-iv) In the case of $\phi \in {C}_{\mathrm{sm}}^{+}\left(I\right)$ and $\psi \in {C}_{\mathrm{sm}}^{-}\left(I\right)$, we find that $\phi \circ {\psi }^{-1}$ is decreasing and concave on $\psi \left(I\right)$ because $\phi ⪯\psi$. Then $-\phi ,\psi \in {C}_{\mathrm{sm}}^{-}\left(I\right)$ and $-\phi \circ {\psi }^{-1}$ is increasing and convex on $\psi \left(I\right)$. Therefore we have from (I-iii) and (2)
${N}_{\left[I,f\right]}\left(\stackrel{˜}{\phi }\right)={N}_{\left[I,f\right]}\left(\stackrel{˜}{-\phi }\right)\le {N}_{\left[I,f\right]}\left(\stackrel{˜}{\psi }\right).$

This completes the proof. □

Remark Let $\phi ,\psi \in {C}_{\mathrm{sm}}\left(I\right)$. We see from Theorem 1 and Lemma 1 that $\psi ⪯\phi$ and then ${N}_{\left[I,f\right]}\left(\stackrel{˜}{\phi }\right)\ge {N}_{\left[I,f\right]}\left(\stackrel{˜}{\psi }\right)$ if any of the following four conditions holds:
1. (v)

$\phi ,\psi \in {C}_{\mathrm{sm}}^{+}\left(I\right)$ and $\phi \circ {\psi }^{-1}$ is convex on $\psi \left(I\right)$.

2. (vi)

$\phi \in {C}_{\mathrm{sm}}^{-}\left(I\right)$, $\psi \in {C}_{\mathrm{sm}}^{+}\left(I\right)$, and $\phi \circ {\psi }^{-1}$ is concave on $\psi \left(I\right)$.

3. (vii)

$\phi ,\psi \in {C}_{\mathrm{sm}}^{-}\left(I\right)$ and $\phi \circ {\psi }^{-1}$ is concave on $\psi \left(I\right)$.

4. (viii)

$\phi \in {C}_{\mathrm{sm}}^{+}\left(I\right)$, $\psi \in {C}_{\mathrm{sm}}^{-}\left(I\right)$, and $\phi \circ {\psi }^{-1}$ is convex on $\psi \left(I\right)$.

Throughout the remainder of the paper, we assume that $\mathrm{\Omega }=I$ and $f\left(x\right)=x$ for all $x\in I$.

Proof of Corollary 1 Let $\phi ,\psi \in {C}_{\mathrm{sm}}\left(I\right)$ with $\phi ⪯\psi$ and ${t}_{1},\dots ,{t}_{n}\in \mathbf{R}$ with ${\sum }_{i=1}^{n}{t}_{i}=1$, $0<{t}_{1}$, ${t}_{n}<1$ and ${t}_{2},\dots ,{t}_{n-1}<0$. Let ${x}_{1},\dots ,{x}_{n}\in I$ be such that ${x}_{1}\le {x}_{2},\dots ,{x}_{n-1}\le {x}_{n}$. Put ${s}_{1}=1-{t}_{1},{s}_{2}=-{t}_{2},\dots ,{s}_{n-1}=-{t}_{n-1},{s}_{n}=1-{t}_{n}$. Then we have ${\sum }_{i=1}^{n}{s}_{i}=1$ and ${s}_{1},\dots ,{s}_{n}>0$. So
$\mu \equiv {s}_{1}{\delta }_{{x}_{1}}+\cdots +{s}_{n}{\delta }_{{x}_{n}}$
is a probability measure on I, where ${\delta }_{x}$ denotes the Dirac measure at $x\in I$. Taking $\left[{x}_{1},{x}_{n}\right]$ instead of I in Theorem 1, we obtain
${\phi }^{-1}\left(\phi \left({x}_{1}\right)+\phi \left({x}_{n}\right)-\sum _{i=1}^{n}{s}_{i}\phi \left({x}_{i}\right)\right)\le {\psi }^{-1}\left(\psi \left({x}_{1}\right)+\psi \left({x}_{n}\right)-\sum _{i=1}^{n}{s}_{i}\psi \left({x}_{i}\right)\right),$
which implies the desired inequality
${\phi }^{-1}\left(\sum _{i=1}^{n}{t}_{i}\phi \left({x}_{i}\right)\right)\le {\psi }^{-1}\left(\sum _{i=1}^{n}{t}_{i}\psi \left({x}_{i}\right)\right).$

This completes the proof. □

Remark Let φ, ψ be in ${C}_{\mathrm{sm}}\left(I\right)$ such that any of (v), (vi), (vii), and (viii) holds. Then $\psi ⪯\phi$ holds from Lemma 1. Therefore if ${t}_{1},\dots ,{t}_{n}\in \mathbf{R}$ with ${\sum }_{i=1}^{n}{t}_{i}=1$, $0<{t}_{1}$, ${t}_{n}<1$, and ${t}_{2},\dots ,{t}_{n-1}<0$, then
${\phi }^{-1}\left(\sum _{i=1}^{n}{t}_{i}\phi \left({x}_{i}\right)\right)\ge {\psi }^{-1}\left(\sum _{i=1}^{n}{t}_{i}\psi \left({x}_{i}\right)\right)$

holds from Corollary 1.

Example 1 Put $\phi \left(x\right)=logx$ and $\psi \left(x\right)=x$ for each positive number $x>0$. Then Corollary 1 easily implies that
$\prod _{i=1}^{n}{x}_{i}^{{t}_{i}}\le \sum _{i=1}^{n}{t}_{i}{x}_{i}$

holds for all ${t}_{1},\dots ,{t}_{n}\in \mathbf{R}$ with ${\sum }_{i=1}^{n}{t}_{i}=1$, $0<{t}_{1}$, ${t}_{n}<1$, and ${t}_{2},\dots ,{t}_{n-1}<0$, and all positive numbers ${x}_{1},\dots ,{x}_{n}$ with ${x}_{1}\le {x}_{2},\dots ,{x}_{n-1}\le {x}_{n}$. This is a geometric-arithmetic mean inequality with negative weights.

Example 2 Put $\phi \left(x\right)=\frac{1}{x}$ and $\psi \left(x\right)=logx$ for each positive number $x>0$. Then Corollary 1 easily implies that
${\left(\sum _{i=1}^{n}\frac{{t}_{i}}{{x}_{i}}\right)}^{-1}\le \prod _{i=1}^{n}{x}_{i}^{{t}_{i}}$

holds for all ${t}_{1},\dots ,{t}_{n}\in \mathbf{R}$ with ${\sum }_{i=1}^{n}{t}_{i}=1$, $0<{t}_{1}$, ${t}_{n}<1$, and ${t}_{2},\dots ,{t}_{n-1}<0$, and all positive numbers ${x}_{1},\dots ,{x}_{n}$ with ${x}_{1}\le {x}_{2},\dots ,{x}_{n-1}\le {x}_{n}$. This is a harmonic-geometric mean inequality with negative weights.

## Declarations

### Acknowledgements

The authors are grateful to the referees for careful reading of the paper and for helpful suggestions and comments. The second author is partially supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science.

## Authors’ Affiliations

(1)
The Open University of Japan, Chiba 261-8586, Japan
(2)
Toho University, Funabashi 273-0866, Japan

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