# Questions tagged [monotone-functions]

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. This tag may also include questions about applications or consequences of monotonicity, such as convergence, optimization, or inequalities.

1,238 questions
Filter by
Sorted by
Tagged with
22 views

### Monotone functions on $\mathbb{R}^n$

I am looking for references on the following topic. It is a known fact that, every measurable function $f:\mathbb{R} \to \mathbb{R}$ monotone can be seen as the sum $f = f_c + f_j$ where $f_c$ is a ...
16 views

### $(n + 1)$-cyclical monotonicity implies cyclical monotonicity in $\mathbb{R}^n$?

Say a function $f : \mathbb{R}^n \to \mathbb{R}^n$ is $N$-cyclically monotone if for any $x_1, \dots, x_{N + 1} \in \mathbb{R}^n$ with $x_1 = x_{N + 1},$ it holds that \begin{align} \sum_{i = 1}^{...
• 155
21 views

• 6,946
33 views

### Find necessary and sufficient conditions for ordinal monotonicity.

First of all let's we remember the following result. Theorem Let be $\lambda$ and ordinal: a predicate $\mathbf P$ is true for any $\alpha$ in $\lambda$ when the truth of $\mathbf P$ for any $\beta$ ...
12 views

### Direction changes in product of functions

Suppose $f(x) = g(x) h(x)$ and the number of direction changes in both $g$ and $h$ are known. Is there a simple way to prove that the number of direction changes in $f$ is bounded by the number of ...
68 views

### Is monotonicity hold of a continuous vector function across union of sets? [closed]

A vector function $F$ is monotonic in $S_i$ if $$(F(x)-F(y))^T (x-y)\geq 0, \forall x,y\in S_i.$$ However, I am unsure if this property holds for continuous functions when considering the union of ...
107 views

### Find all $x \in \mathbb{R}$ such that $f: \mathbb{N} \rightarrow \mathbb{R}$, where $f(n) = \{2^n \cdot x\}$ is monotone on $\mathbb{N}$.

Find all $x \in \mathbb{R}$ such that $f: \mathbb{N} \rightarrow \mathbb{R}$, where $f(n) = \{2^n \cdot x\}$ (where $\{x\}$ denotes the fractional part of $x$), is increasing/decreasing on $\mathbb{N}$...
33 views

### A question about the monotonicity of $y_1 = x^e$ and $y_2 = x^{\pi}$

As we know from 1st year Calculus, for function $f(x) = x^n, x \in \mathbb{R}, n \in \mathbb{N}$, if $n$ is odd, the $f(x)$ is increasing; if $n$ is even $f(x)$ is increasing when $x \geq 0$ and ...
• 1,164
9 views

### Clamp endpoint derivatives of cubic polynomail such that the polynomial becomes monotonic on interval

I have a polynomial $P(x) = Ax^3 + Bx^2 + Cx + D$, with $P(0) = 1$ and $P(1) = 0$. This means that $D=1$ and $A + B + C + D = 0$. Suppose that $P'(0) = d_0 \leq 0$ and $P'(1) = d_1 \leq 0$. Depending ...
• 231
38 views

### Classifing functions with the property that the preimage of path-connected sets is path-connected

Given two path-connected topological spaces $\mathcal{X},\mathcal{Y}$, consider a function $f: \mathcal{X} \to \mathcal{Y}$ with the property, that the preimage $f^{-1}(V)$ is pathconnected for all ...
1 vote
152 views

### How to prove a function is strictly increasing?

Let $f(x)$ be a continuous strictly increasing smooth function on $[0,1]$ with $f(0)=0$ and $f(1)=1$. In addition, $F(t)=\int_0^t f(x)dx$ and $\phi>1$. Prove that the following function $h(t)$ is ...
• 35
63 views

### Proving the sequence of harmonic function uniformly converges on compact sets.

Fix an open, bounded $U \subset \mathbb{R}^n$ and consider a sequence $C^2(U) \cap C(\bar{U})$ of harmonic functions. (i) Suppose that $u_n$ converges uniformly to a function $u.$ Show that u is ...
1 vote
22 views

### Monotonicity of the probability of a sum of independent random variables being below a threshold

Suppose I have a sequence of i.i.d. random variables $X_1, X_2, \dots$ with positive mean $E[X_1] = \mu$. For $A>0$, is the function $$f(n) = \Pr( X_1 + X_2 + \dots + X_n \leq A)$$ monotonically ...
• 31
23 views

### When is a sequence that is bounded below by an unbounded and monotone increasing also monotone increasing? [closed]

Let $(a_n)$ be an unbounded and monotone increasing sequence of positive integers and let $(b_n)$ be another sequence of positive integers such that for all $n \in \mathbb{N}$, $a_n \leq b_n$. ...
88 views

### $x^n/(1+x^n)$is monotonic

$x^n/(1+x^n)$ is monotonic , $x \in R^+$ , $n \in N$ tried solving it by differentiating but not getting feasible result, also is there any other way of computation through which this can be solved ...
73 views

### Number of monotonically increasing functions such that $f(i)\le i$.

Problem: Consider $n \in \mathbb{N}^+$, set $A = \mathbb{N}^+ _{\leq n}$. Find the number of monotonically increasing functions $f: A → A$ such that $f(i) \leq i$. I tried using the multinomial ...
28 views

• 4,468
50 views

### Concave continous function with special property implies that is strictly increasing?

I have the following problem. Let $\rho(s)\geq 0$ for all $s\geq 0$ be a continuous concave function, satisfying $\rho(s)\geq\rho_1(s)$ for all $s>0$, where $\rho_1$ is a strictly increasing and ...
69 views

### Show that the recursive sequence $x_{k+1} = |x_k - \frac{x_k}{1-2M^2x_k^2}|$ is monotone

I'm doing some exercises for an upcoming exam, and as part of a larger problem, I want to show that the given recursive sequence: $$x_{k+1} = \left|x_k - \frac{x_k}{1-2M^2x_k^2}\right|$$ is monotonly ...
• 105
1 vote
43 views

• 2,016
114 views

### Equivalent condition for the following monotone-like condition for multivariate function

Let $f:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb R$ be a continuously twice differentiable function. I am considering the following condition (1) on $f$: For any $x,x'\in\mathbb R^n$, If it is ...
1 vote
110 views

• 26.8k
28 views

• 4,468
199 views

### Monotonicity of a multivariate function

Let $f:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb R$. Assume $f(x,y)$ is twise continuously differentiable and the derivative with respect to both arguments (i.e., $x$ and $y$) are monotone in ...
1 vote
Let $f:\mathbb R^n\times\mathbb R^n\rightarrow \mathbb R$ be a twice continuously differentiable function. The cross-derivative of $f$ is denoted by $D_{xy}f$. I want to check whether the following ...