- Research Article
- Open Access

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

- Yasuo Nakasuji
^{1}Email author and - Sin-Ei Takahasi
^{2}

**2014**:450

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

© Nakasuji and Takahasi; licensee Springer. 2014

**Received:**23 June 2014**Accepted:**21 October 2014**Published:**6 November 2014

## 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].

*I*be a finite closed interval $[m,M]$ on

**R**and $C(I)$ the space of all continuous real-valued functions defined on

*I*. Moreover, let ${C}_{\mathrm{sm}}^{+}(I)$ (resp. ${C}_{\mathrm{sm}}^{-}(I)$) be the set of all functions in $C(I)$ which are strictly monotone increasing (resp. decreasing) on

*I*. Put

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

*f*a function in ${L}^{1}(\mathrm{\Omega},\mu )$ such that $f(\omega )\in I$ for almost all $\omega \in \mathrm{\Omega}$. Then we see that $\phi \circ f\in {L}^{1}(\mathrm{\Omega},\mu )$ for all $\phi \in {C}_{\mathrm{sm}}(I)$ because

*φ*is a bounded continuous function and

*μ*is a finite measure. Put

for each $\phi \in {C}_{\mathrm{sm}}(I)$. Then [[1], Theorem 1] which gives a new interpretation of Jensen’s inequality is restated as $\tilde{\phi}\u2aaf\tilde{\psi}\Rightarrow {M}_{\phi}(f)\le {M}_{\psi}(f)$. In this paper, we give a new order-preserving average function ${N}_{[I,f]}$ on the quotient space ${\tilde{C}}_{\mathrm{sm}}(I)$, according to this idea. We also apply this function ${N}_{[I,f]}$ to derive a finite form of Jensen type inequality with negative weights.

*φ*be an arbitrary function of ${C}_{\mathrm{sm}}(I)$. Since $\phi (I)$ is an interval of

**R**and

*μ*is a probability measure on Ω, it follows that

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

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

**Theorem 1** ${N}_{[I,f]}$ *is an order*-*preserving real*-*valued function on the quotient space* ${\tilde{C}}_{\mathrm{sm}}(I)$ *with order* ⪯, *that is*, $\tilde{\phi}\u2aaf\tilde{\psi}\Rightarrow {N}_{[I,f]}(\tilde{\phi})\le {N}_{[I,f]}(\tilde{\psi})$.

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

**Corollary 1**

*Let*$\phi ,\psi \in {C}_{\mathrm{sm}}(I)$

*with*$\tilde{\phi}\u2aaf\tilde{\psi}$

*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*

*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 ${\tilde{C}}_{\mathrm{sm}}(I)$

Let us start with the following two lemmas.

**Lemma 1**

*Let*$\phi \in {C}_{\mathrm{sm}}(I)$.

*Then*:

- (i)
*φ**is increasing and convex on**I**if and only if*${\phi}^{-1}$*is increasing and concave on*$\phi (I)$. - (ii)
*φ**is increasing and concave on**I**if and only if*${\phi}^{-1}$*is increasing and convex on*$\phi (I)$. - (iii)
*φ**is decreasing and convex on**I**if and only if*${\phi}^{-1}$*is decreasing and convex on*$\phi (I)$. - (iv)
*φ**is decreasing and concave on**I**if and only if*${\phi}^{-1}$*is decreasing and concave on*$\phi (I)$.

*Proof* Straightforward. □

**Lemma 2**

- (i)
*If**φ**is a convex function on**I**and**ψ**is an increasing convex function on*$\phi (I)$,*then*$\psi \circ \phi $*is convex on**I*. - (ii)
*If**φ**is a convex function on**I**and**ψ**is a decreasing concave function on*$\phi (I)$,*then*$\psi \circ \phi $*is concave on**I*. - (iii)
*If**φ**is a concave function on**I**and**ψ**is an increasing concave function on*$\phi (I)$,*then*$\psi \circ \phi $*is concave on**I*. - (iv)
*If**φ**is a concave function on**I**and**ψ**is a decreasing convex function on*$\phi (I)$,*then*$\psi \circ \phi $*is convex on**I*.

*Proof* Straightforward. □

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

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

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

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

**Remark** Lemma 1 guarantees that the above $\phi \u2aaf\psi $ is a restatement of the concepts appearing in [[1], Lemma 3].

**Lemma 3** *Let* $\phi ,{\phi}^{\prime},\psi ,{\psi}^{\prime}\in {C}_{\mathrm{sm}}(I)$. *If* $\phi \cong {\phi}^{\prime}$, $\psi \cong {\psi}^{\prime}$, *and* $\phi \u2aaf\psi $, *then* ${\phi}^{\prime}\u2aaf{\psi}^{\prime}$.

*Proof*Assume that $\phi \cong {\phi}^{\prime}$, $\psi \cong {\psi}^{\prime}$, and $\phi \u2aaf\psi $. Then we must show ${\phi}^{\prime}\u2aaf{\psi}^{\prime}$. Since $\phi \cong {\phi}^{\prime}$, $\psi \cong {\psi}^{\prime}$, we can write ${\phi}^{\prime}$ and ${\psi}^{\prime}$ as follows:

□

For any $\tilde{\phi},\tilde{\psi}\in {\tilde{C}}_{\mathrm{sm}}(I)$, we write $\tilde{\phi}\u2aaf\tilde{\psi}$ by the same notation if $\phi \u2aaf\psi $ holds. This is well defined by Lemma 3. In this case, we have the following.

**Theorem 2** ⪯ *is an order relation in* ${\tilde{C}}_{\mathrm{sm}}(I)$.

*Proof*We show the theorem by dividing into three steps.

- (I)
It is evident that ⪯ satisfies the reflexivity.

- (II)Assume that $\tilde{\phi}\u2aaf\tilde{\psi}$ and $\tilde{\psi}\u2aaf\tilde{\phi}$. Then $\phi \u2aaf\psi $ and $\psi \u2aaf\phi $ hold. In the case of $\phi ,\psi \in {C}_{\mathrm{sm}}^{+}(I)$, we find that $\phi \circ {\psi}^{-1}$ is concave on $\psi (I)$ and $\psi \circ {\phi}^{-1}$ is concave on $\phi (I)$. Since $\psi \circ {\phi}^{-1}$ is increasing and concave on $\phi (I)$, it follows from Lemma 1-(ii) that $\phi \circ {\psi}^{-1}={(\psi \circ {\phi}^{-1})}^{-1}$ is convex on $\psi (I)$. Therefore $\phi \circ {\psi}^{-1}$ is affine on $\psi (I)$ and hence $\phi \cong \psi $, that is, $\tilde{\phi}=\tilde{\psi}$. By the same method, we can easily see that $\tilde{\phi}=\tilde{\psi}$ holds in the other three cases:$[\phi \in {C}_{\mathrm{sm}}^{+}(I),\psi \in {C}_{\mathrm{sm}}^{-}(I)],\phantom{\rule{2em}{0ex}}[\phi \in {C}_{\mathrm{sm}}^{-}(I),\psi \in {C}_{\mathrm{sm}}^{+}(I)]\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}[\phi ,\psi \in {C}_{\mathrm{sm}}^{-}(I)].$

- (III)Assume that $\tilde{\phi}\u2aaf\tilde{\psi}$ and $\tilde{\psi}\u2aaf\tilde{\lambda}$. Then $\phi \u2aaf\psi $ and $\psi \u2aaf\lambda $ hold. In the case of $\phi ,\psi ,\lambda \in {C}_{\mathrm{sm}}^{+}(I)$, we find that $\phi \circ {\psi}^{-1}$ is increasing and concave on $\psi (I)$ and $\psi \circ {\lambda}^{-1}$ is concave on $\lambda (I)$. Then it follows from Lemma 2-(iii) that $\phi \circ {\lambda}^{-1}=(\phi \circ {\psi}^{-1})\circ (\psi \circ {\lambda}^{-1})$ is concave on $\lambda (I)$, and hence $\phi \u2aaf\lambda $, that is, $\tilde{\phi}\u2aaf\tilde{\lambda}$ holds. By the same method, we can easily see that $\tilde{\phi}\u2aaf\tilde{\lambda}$ holds in the other seven cases:$\begin{array}{c}[\phi \in {C}_{\mathrm{sm}}^{+}(I),\psi \in {C}_{\mathrm{sm}}^{+}(I),\lambda \in {C}_{\mathrm{sm}}^{-}(I)],\phantom{\rule{2em}{0ex}}\dots ,\hfill \\ [\phi \in {C}_{\mathrm{sm}}^{-}(I),\psi \in {C}_{\mathrm{sm}}^{-}(I),\lambda \in {C}_{\mathrm{sm}}^{-}(I)].\hfill \end{array}$

Therefore ⪯ satisfies the transitive law. □

## 3 Proofs of Theorem 1 and Corollary 1

*φ*be an arbitrary function of ${C}_{\mathrm{sm}}(I)$. Then an easy observation implies that

**Lemma 4**

*Let*$\phi \in {C}_{\mathrm{sm}}(I)$.

*If either*

*φ*

*is increasing and concave on*

*I*

*or decreasing and convex on*

*I*,

*then*

*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 $\phi (I)$ by Lemma 1-(ii), and hence the first inequality in Lemma 4 follows from Jensen’s inequality. Put

*I*such that ${\phi}^{\mathrm{\u266f}}(m)={\phi}^{\mathrm{\u266f}}(M)=m+M$. Therefore we have

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 $\phi (I)$ by Lemma 1-(i), and hence the first inequality in Lemma 4 is reversed from Jensen’s inequality. Also since ${\phi}^{\mathrm{\u266f}}$ 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 $\tilde{\phi},\tilde{\psi}\in {\tilde{C}}_{\mathrm{sm}}(I)$ with $\tilde{\phi}\u2aaf\tilde{\psi}$, where $\phi ,\psi \in {C}_{\mathrm{sm}}(I)$.

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

This completes the proof. □

**Remark**Let $\phi ,\psi \in {C}_{\mathrm{sm}}(I)$. We see from Theorem 1 and Lemma 1 that $\psi \u2aaf\phi $ and then ${N}_{[I,f]}(\tilde{\phi})\ge {N}_{[I,f]}(\tilde{\psi})$ if any of the following four conditions holds:

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

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

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

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

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

*Proof of Corollary 1*Let $\phi ,\psi \in {C}_{\mathrm{sm}}(I)$ with $\phi \u2aaf\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

*I*, where ${\delta}_{x}$ denotes the Dirac measure at $x\in I$. Taking $[{x}_{1},{x}_{n}]$ instead of

*I*in Theorem 1, we obtain

This completes the proof. □

**Remark**Let

*φ*,

*ψ*be in ${C}_{\mathrm{sm}}(I)$ such that any of (v), (vi), (vii), and (viii) holds. Then $\psi \u2aaf\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

holds from Corollary 1.

**Example 1**Put $\phi (x)=logx$ and $\psi (x)=x$ for each positive number $x>0$. Then Corollary 1 easily implies that

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 $\phi (x)=\frac{1}{x}$ and $\psi (x)=logx$ for each positive number $x>0$. Then Corollary 1 easily implies that

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

## References

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