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

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.

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 $[m,M]$ 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

$$C_{\mathrm{sm}}(I)=C_{\mathrm{sm}}^{+}(I)\cup C_{\mathrm{sm}}^{-}(I).$$

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).

Let $(\mathrm{\Omega },\mu )$ be a probability space and 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

$$M_{\phi }(f)={\phi }^{-1}(\int \phi \circ fd\mu )$$

for each $\phi \in C_{\mathrm{sm}}(I)$. Then [[1], Theorem 1] which gives a new interpretation of Jensen’s inequality is restated as $\tilde{\phi }⪯\tilde{\psi }\Rightarrow M_{\phi }(f)\le M_{\psi }(f)$. 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.

Let 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

$$\phi (m)+\phi (M)-\int \phi \circ fd\mu \in \phi (I),$$

and hence we have

$${\phi }^{-1}(\phi (m)+\phi (M)-\int \phi \circ fd\mu )\in I.$$

Note that a simple computation implies that if $\phi ,\psi \in C_{\mathrm{sm}}(I)$ satisfy $\tilde{\phi }=\tilde{\psi }$, then

$${\phi }^{-1}(\phi (m)+\phi (M)-\int \phi \circ fd\mu )={\psi }^{-1}(\psi (m)+\psi (M)-\int \psi \circ fd\mu )$$

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, $\tilde{\phi }⪯\tilde{\psi }\Rightarrow N_{[I,f]}(\tilde{\phi })\le N_{[I,f]}(\tilde{\psi })$.

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 $\tilde{\phi }⪯\tilde{\psi }$ 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

$${\phi }^{-1}(\sum _{i=1}^n{t}_i\phi (x_i))\le {\psi }^{-1}(\sum _{i=1}^n{t}_i\psi (x_i))$$

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. □

For any $\phi ,\psi \in C_{\mathrm{sm}}(I)$, we write $\phi ⪯\psi$ if any of the following four conditions holds:

• (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 $\phi ⪯\psi$ 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 $\phi ⪯\psi$, 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:

$${\phi }^{\prime }=a\phi +b\text{and}{\psi }^{\prime }=c\psi +d$$

for some $a,b,c,d\in \mathbf{R}$. Then we have $a\ne 0$ and $c\ne 0$. Put

$$\zeta (x)=ax+b\text{and}\eta (x)=cx+d$$

for each $x\in \mathbf{R}$. In the case of $\phi ,\psi \in C_{\mathrm{sm}}^{+}(I)$ and $a,c>0$, we find that $\phi \circ {\psi }^{-1}$ is concave on $\psi (I)$ because $\phi ⪯\psi$. Then $\zeta \circ \phi \circ {\psi }^{-1}$ is increasing and concave on $\psi (I)$ 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 ${\psi }^{\prime }(I)$ from Lemma 2-(iii). However, since ${\phi }^{\prime },{\psi }^{\prime }\in C_{\mathrm{sm}}^{+}(I)$, we obtain ${\phi }^{\prime }⪯{\psi }^{\prime }$ as required. Moreover, we can easily see that ${\phi }^{\prime }⪯{\psi }^{\prime }$ holds in the other 15 cases:

$$\begin{array}{c}[\phi \in C_{\mathrm{sm}}^{+}(I),\psi \in C_{\mathrm{sm}}^{-}(I),a>0,c>0],\dots ,\hfill \\ [\phi \in C_{\mathrm{sm}}^{-}(I),\psi \in C_{\mathrm{sm}}^{-}(I),a<0,c<0].\hfill \end{array}$$

 □

For any $\tilde{\phi },\tilde{\psi }\in {\tilde{C}}_{\mathrm{sm}}(I)$, we write $\tilde{\phi }⪯\tilde{\psi }$ 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 ${\tilde{C}}_{\mathrm{sm}}(I)$.

Proof We show the theorem by dividing into three steps.

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

• (II) Assume that $\tilde{\phi }⪯\tilde{\psi }$ and $\tilde{\psi }⪯\tilde{\phi }$. Then $\phi ⪯\psi$ and $\psi ⪯\phi$ 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:

  $$[\phi \in C_{\mathrm{sm}}^{+}(I),\psi \in C_{\mathrm{sm}}^{-}(I)],[\phi \in C_{\mathrm{sm}}^{-}(I),\psi \in C_{\mathrm{sm}}^{+}(I)]\text{and}[\phi ,\psi \in C_{\mathrm{sm}}^{-}(I)].$$

Therefore ⪯ satisfies the symmetry law.

• (III) Assume that $\tilde{\phi }⪯\tilde{\psi }$ and $\tilde{\psi }⪯\tilde{\lambda }$. Then $\phi ⪯\psi$ and $\psi ⪯\lambda$ 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 $\phi ⪯\lambda$, that is, $\tilde{\phi }⪯\tilde{\lambda }$ holds. By the same method, we can easily see that $\tilde{\phi }⪯\tilde{\lambda }$ holds in the other seven cases:

  $$\begin{array}{c}[\phi \in C_{\mathrm{sm}}^{+}(I),\psi \in C_{\mathrm{sm}}^{+}(I),\lambda \in C_{\mathrm{sm}}^{-}(I)],\dots ,\hfill \\ [\phi \in C_{\mathrm{sm}}^{-}(I),\psi \in C_{\mathrm{sm}}^{-}(I),\lambda \in C_{\mathrm{sm}}^{-}(I)].\hfill \end{array}$$

Therefore ⪯ satisfies the transitive law. □

3 Proofs of Theorem 1 and Corollary 1

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

$${(-\phi )}^{-1}(y)={\phi }^{-1}(-y)$$

(1)

for all $y\in -\phi (I)$ and that

$$N_{[I,f]}(\widetilde{-\phi })=N_{[I,f]}(\tilde{\phi }).$$

(2)

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

$$N_{[I,f]}(\tilde{\phi })\le \int {\phi }^{-1}\circ (\phi (m)+\phi (M)-\phi \circ f)d\mu \le m+M-\int fd\mu$$

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

$${\phi }^{\mathrm{♯}}(x)={\phi }^{-1}(\phi (m)+\phi (M)-\phi (x))+x$$

for each $x\in I$. Then it follows from Lemma 2-(i) that ${\phi }^{\mathrm{♯}}$ is a convex function on I such that ${\phi }^{\mathrm{♯}}(m)={\phi }^{\mathrm{♯}}(M)=m+M$. Therefore we have

$${\phi }^{-1}(\phi (m)+\phi (M)-\phi (f(\omega )))\le m+M-f(\omega )$$

(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 $\phi (I)$ 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 $\tilde{\phi },\tilde{\psi }\in {\tilde{C}}_{\mathrm{sm}}(I)$ with $\tilde{\phi }⪯\tilde{\psi }$, where $\phi ,\psi \in C_{\mathrm{sm}}(I)$.

(I-i) In the case of $\phi ,\psi \in C_{\mathrm{sm}}^{+}(I)$, we find that $\phi \circ {\psi }^{-1}$ is increasing and concave on $\psi (I)=[\psi (m),\psi (M)]$ because $\phi ⪯\psi$. Therefore we have from Lemma 4

$$\begin{array}{c}\psi (N_{[I,f]}(\tilde{\phi }))\hfill \\ =(\psi \circ {\phi }^{-1})(\phi (m)+\phi (M)-\int \phi \circ fd\mu )\hfill \\ ={(\phi \circ {\psi }^{-1})}^{-1}((\phi \circ {\psi }^{-1})(\psi (m))+(\phi \circ {\psi }^{-1})(\psi (M))-\int (\phi \circ {\psi }^{-1})\circ (\psi \circ f)d\mu )\hfill \\ =N_{[\psi (I),\psi \circ f]}\left(\widetilde{\phi \circ {\psi }^{-1}}\right)\hfill \\ \le \psi (m)+\psi (M)-\int \psi \circ fd\mu \hfill \\ =\psi (N_{[I,f]}(\tilde{\psi })),\hfill \end{array}$$

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

(I-ii) In the case of $\phi \in C_{\mathrm{sm}}^{-}(I)$ and $\psi \in C_{\mathrm{sm}}^{+}(I)$, we find that $\phi \circ {\psi }^{-1}$ is decreasing and convex on $\psi (I)$ because $\phi ⪯\psi$. Then $-\phi ,\psi \in C_{\mathrm{sm}}^{+}(I)$ and $(-\phi )\circ {\psi }^{-1}$ is increasing and concave on $\psi (I)$. Therefore we have from (I-i) and (2)

$$N_{[I,f]}(\tilde{\phi })=N_{[I,f]}(\widetilde{-\phi })\le N_{[I,f]}(\tilde{\psi }).$$

(I-iii) In the case of $\phi ,\psi \in C_{\mathrm{sm}}^{-}(I)$, we find that $\phi \circ {\psi }^{-1}$ is increasing and convex on $\psi (I)$ because $\phi ⪯\psi$. Then $\phi \in C_{\mathrm{sm}}^{-}(I)$, $-\psi \in C_{\mathrm{sm}}^{+}(I)$, and $\phi \circ {(-\psi )}^{-1}$ is decreasing and convex on $-\psi (I)$ by (1). Therefore we have from (I-ii) and (2)

$$N_{[I,f]}(\tilde{\phi })\le N_{[I,f]}(\widetilde{-\psi })=N_{[I,f]}(\tilde{\psi }).$$

(I-iv) In the case of $\phi \in C_{\mathrm{sm}}^{+}(I)$ and $\psi \in C_{\mathrm{sm}}^{-}(I)$, we find that $\phi \circ {\psi }^{-1}$ is decreasing and concave on $\psi (I)$ because $\phi ⪯\psi$. Then $-\phi ,\psi \in C_{\mathrm{sm}}^{-}(I)$ and $-\phi \circ {\psi }^{-1}$ is increasing and convex on $\psi (I)$. Therefore we have from (I-iii) and (2)

$$N_{[I,f]}(\tilde{\phi })=N_{[I,f]}(\widetilde{-\phi })\le N_{[I,f]}(\tilde{\psi }).$$

This completes the proof. □

Remark Let $\phi ,\psi \in C_{\mathrm{sm}}(I)$. We see from Theorem 1 and Lemma 1 that $\psi ⪯\phi$ 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 $\phi ⪯\psi$ 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 $[x_1,x_n]$ instead of I in Theorem 1, we obtain

$${\phi }^{-1}(\phi (x_1)+\phi (x_n)-\sum _{i=1}^n{s}_i\phi (x_i))\le {\psi }^{-1}(\psi (x_1)+\psi (x_n)-\sum _{i=1}^n{s}_i\psi (x_i)),$$

which implies the desired inequality

$${\phi }^{-1}(\sum _{i=1}^n{t}_i\phi (x_i))\le {\psi }^{-1}(\sum _{i=1}^n{t}_i\psi (x_i)).$$

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 $\psi ⪯\phi$ 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

$${\phi }^{-1}(\sum _{i=1}^n{t}_i\phi (x_i))\ge {\psi }^{-1}(\sum _{i=1}^n{t}_i\psi (x_i))$$

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

$$\prod _{i=1}^n{x}_i^{t_i}\le \sum _{i=1}^n{t}_i{x}_i$$

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 $\phi (x)=\frac{1}{x}$ and $\psi (x)=logx$ 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.