CAL1_L12

Monotonicity and Bimonotonicity

Definition 1(Definition).

Let $D\subseteq {\mathbb{R}}^n$ and let $f:D\to \mathbb{R}$ be any function. Let $I$ and $J$ be intervals in $\mathbb{R}$ such that $I\times J\subseteq D$. We say that

1. $f$ is monotonically increasing on $I\times J$ if for all $\mathbf{x},\mathbf{y}\in I\times J$, $\mathbf{x}\le \mathbf{y}⟹f(\mathbf{x})\le f(\mathbf{y})$.
2. $f$ is monotonically decreasing on $I\times J$ if for all $\mathbf{x},\mathbf{y}\in I\times J$, $\mathbf{x}\le \mathbf{y}⟹f(\mathbf{x})\ge f(\mathbf{y})$.
3. $f$ is bimonotonically increasing on $I\times J$ if for all $\mathbf{x},\mathbf{y}\in I\times J$, $\mathbf{x}\le \mathbf{y}⟹$ $f(x_1,y_2)+f(x_2,y_1)\le f(x_1,y_1)+f(x_2,y_2)$.
4. $f$ is bimonotonically decreasing on $I\times J$ if for all $\mathbf{x},\mathbf{y}\in I\times J$, $\mathbf{x}\le \mathbf{y}⟹$ $f(x_1,y_2)+f(x_2,y_1)\ge f(x_1,y_1)+f(x_2,y_2)$.
5. $f$ is bimonotonic on $I\times J$ if $f$ is bimonotonically increasing or bimonotonically decreasing on $I\times J$.

Theorem 2(Proposition).

Let $I,J$ be nonempty intervals in $\mathbb{R}$. Given any $\varphi :I\to \mathbb{R}$ and $\psi :I\to \mathbb{R}$, consider $f:I\times J\to \mathbb{R}$ and $g:I\times J\to \mathbb{R}$ defined by

$$f(x,y)\equiv \varphi (x)+\psi (y),$$ $$g(x,y)\equiv \varphi (x)\psi (y)$$

for $(x,y)\in I\times J$. Then,

1. $f$ is monotonically increasing on $I\times J$ iff $\varphi$ is increasing on $I$ and $\psi$ is increasing on $J$.
2. Assume $\varphi (x)\ge 0$ and $\psi (y)\ge 0$ for all $x\in I$ and $y\in J$, and $\varphi (x_0)>0$ and $\psi (y_0)>0$ for some $x_0\in I$ and $y_0\in J$. Then $g$ is monotonically increasing on $I\times J$ iff $\varphi$ is increasing on $I$ and $\psi$ is increasing on $J$.
3. $f$ is always bimonotonically increasing and decreasing on $I\times J$.
4. If $\varphi$ is monotonic on $I$ and $\psi$ is monotonic on $J$, then $g$ is bimonotonic on $I\times J$. More specifically, if $\varphi$ and $\psi$ are both increasing or both decreasing, then $g$ is bimonotonically increasing, whereas if one of them is increasing and the other is decreasing, then $g$ is bimonotonically decreasing.

Info

A function $f:I\to \mathbb{R}$ where $I\subseteq \mathbb{R}$ is an interval is said to be concave if for any $\alpha \in [0,1]$,

$$f((1-\alpha )x+\alpha y)\ge (1-\alpha )f(x)+\alpha f(y).$$

Similar definition for convex functions.

Theorem 3(Proposition).

Let $I,J$ be nonempty intervals in $\mathbb{R}$. The set

$$I+J\equiv \{x+y|x\in I\text{ and }y\in J\}$$

is an interval in $\mathbb{R}$. Further, let $\varphi :I+J\to \mathbb{R}$ be any function and consider $f:I\times J\to \mathbb{R}$ defined by

$$f(x,y)\equiv \varphi (x+y)$$

for $(x,y)\in I\times J$. Then,

1. $\varphi$ is increasing/decreasing on $I+J$ $⟹$ $f$ is monotonically increasing/decreasing on $I\times J$.

2. $\varphi$ is convex/concave on $I+J$ $⟹$ $f$ is bimonotonically increasing/decreasing on $I\times J$

Proof
First, let’s show that $I+J$ is an interval. Let $x_1,x_2\in I$ and $y_1,y_2\in J$ be such that $x_1+y_1\le x_2+y_2$ and consider $r\in \mathbb{R}$ such that $x_1+y_1<r<x_2+y_2$. Then, there exists $t\in [0,1]$ such that $r=t(x_1+y_1)+(1-t)(x_2+y_2)=$ $t{x}_1+(1-t)x_2+t{y}_1+(1-t)y_2$. Note that $t{x}_1+(1-t)(x_2)\in I$ and $t{y}_1+(1-t)y_2\in J$, and we are done.

$(1)$ should be clear. Next, suppose $\varphi$ is convex on $I+J$. Consider $\mathbf{x},\mathbf{y}\in I\times J$ such that $\mathbf{x}\le \mathbf{y}$ and $\mathbf{x}\ne \mathbf{y}$. Note that

$$x_1+y_2=\lambda (x_1+y_1)+(1-\lambda )(x_2+y_2)$$

and

$$x_2+y_1=(\lambda -1)(x_1+y_1)+\lambda (x_2+y_2),$$

where $\lambda \in [0,1]$ can be explicitly computed. Since $\varphi$ is convex,

$$\varphi (x_1+y_2)\le \lambda \varphi (x_1+y_1)+(1-\lambda )(x_2+y_2)$$

and

$$\varphi (x_2+y_1)\le \lambda \varphi (x_2+y_2)+(1-\lambda )\varphi (x_1+y_1).$$

Thus,

$$\varphi (x_1+y_2)+\varphi (x_2+y_1)\le \varphi (x_1+y_1)+\varphi (x_2+y_2).$$

It follows that $f$ is bimonotonically increasing.