Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

-1
votes
0answers
20 views

prove the Strictly Increasingness of this Function

\begin{eqnarray*} \Gamma(x+1)>\frac {\frac{(900\gamma^2+73\pi^2)x}{900\gamma}+\frac{73}{100}} {(\frac{\pi^2 x}{9\gamma}+1)} \quad \textbf{[2]} \end{eqnarray*} i want to prove it is strictly ...
2
votes
0answers
61 views

Does the convex cone of monotonic functions on a compact set admit a countable conic generating set?

For any function $f:[0,1]^d\rightarrow\mathbb{R}$ and any $i\in\{1,\dots,d\}$ and $x\in[0,1]^d$, let the functions $f_{i,x}:[0,1]\rightarrow\mathbb{R}$ and $\partial_i f:[0,1]^d\rightarrow\mathbb{R}$ ...
1
vote
0answers
21 views

Monotonicity of the expectation of random variables

Define $r(t)=\frac{1+X_1+...+X_t}{1+Y_1+...+Y_t}$ $X_t\sim Binomial(Y_t,\alpha)$ $Y_t \sim Binomial(\lfloor 1+r(t-1)\rfloor,\delta)$ I would like to show that $E[r(t)]$ is a monotonic function of $t$...
4
votes
1answer
92 views

If $f: [1,\infty)\to [e,+\infty)$ is increasing, $\int_1^\infty \frac{dx}{f(x)}=+\infty$, show that $\int_1^\infty \frac{dx}{x\ln f(x)}=+\infty$. [closed]

If $f: [1,\infty)\to [e,+\infty)$ is increasing and $$\int_1^\infty \frac{dx}{f(x)}=+\infty$$show that $$\int_1^\infty \frac{dx}{x\ln f(x)}=+\infty$$ How can I show this?
0
votes
1answer
42 views

Proving the complete monotonicity of $f$ given $(−\log f)′$ is completely monotone

A function $f:(0,∞)→[0,∞]$ is said to be completely monotonic if its $n$-th derivative exists and $(−1)^nf^{(n)}(x)≥0$, where $f^{(n)}(x)$ is the $n$-th derivative of $f$. Prove that if $(−\log f(x))′...
0
votes
0answers
37 views

Cauchy-Schwarz inversion like inequality for expectactions of comonotonic functions

Given two non constant, integrable, comonotonic functions $x_1, x_2\colon [0,\infty) \to [0,1]$, i.e., both functions are non decreasing or non increasing, I need to prove that $$\big(E[x_1(T)]+E[x_2(...
1
vote
0answers
19 views

Does local monotonicity implies global monotonicity?

Let $f:(-\pi, \pi) \longrightarrow \mathbb{R} $ be a function such that for every $x \in (-\pi, \pi) $, $f'(x)$ exists and it's positive. This means that given $x_o \in (-\pi, \pi)$, we have that $\...
0
votes
1answer
26 views

What does this monotonicity condition on the gradient correspond to?

We know that all the gradient of convex functions $f: X \to \mathbb{R}, X$ convex, are monotone maps, that is, $f$ convex, continuously differentiable $\implies (\nabla f(x) - \nabla f(x^\prime)^T(...
1
vote
0answers
47 views

Proof of decreasing sequence [duplicate]

Can someone help me prove that the sequence $$\left(1+\frac{1}{N} \right)^{N+1}$$ is decreasing for $N \in \Bbb N $? EDIT: I have tried the following.. \begin{align} \frac{\left(1+\frac{1}{...
1
vote
1answer
33 views

Does the system $xy = ab, G(x)+G(y)=G(a)+G(b)$ always have exactly two solutions if $G$ is continuous and injective?

If $f$ and $g$ are commutative operations $$\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R},$$ then for any constants $a,b \in \mathbb{R}$, the system of equations $$f(x,y) = f(a,b), \qquad g(x,y) ...
0
votes
2answers
27 views

sequence of monotone measures

I have a sigma-algebra F on X and a monotone sequence of measures on F, meaning: $$ \mu_n(A)\leq \mu_{n+1} (A) \forall A \in F $$ It should hold that $ \mu_1(X) <\infty $ I want to show, that $$\...
1
vote
1answer
41 views

Uniform convergence of $\frac{f(xt)}{f(t)}$ as $t\to\infty$

Let $f(x)$ decrease monotonically to $0$ as $x\to\infty$. Suppose $\lim\limits_{t\to\infty}\frac{f(xt)}{f(t)} = x^p$ for all $x>0$ and some $p<0$. Can this convergence be nonuniform in $x$?
1
vote
0answers
20 views

What does it mean for a pivotal quantity to be monotone in $\theta$?

I'm told that a pivotal quantity is a function $g(\mathbf X,\theta)$ of the data $\mathbf X$ and the parameter $\theta$, satisfying The distribution of $g(\mathbf X,\theta)$ is known and independent ...
0
votes
0answers
36 views

For an increasing function $f$, if $f(y_n)-f(x_n) \to 0$ , then $f$ is continuous at $0$

Question: Let $f: \mathbb{R} \to \mathbb{R} $ be an increasing function. Suppose there are sequences $(x_n)$ and $(y_n)$ such that $x_n<0<y_n$ for all $n\geq 1$ and $f(y_n)-f(x_n) \to 0$ as $n \...
0
votes
1answer
17 views

Validity of a statement in a proof (regarding continuity)

Does the equality $f(a,b)=(f(a),f(b)) $ hold? I doubt it. Taking $f(x)= | x|$ and $a=-5, b=6$ gives us $f(-5,6)= (5,6)$, but $0 \notin (5,6)$. This is indeed a very stupid question, but I am not ...
1
vote
1answer
31 views

Discontinuity of monotonic function

I've seen this topic has been already discussed in this question but actually my doubt is slightly different so I consider opportune to ask it as a sigle question, please correct me if I am wrong. ...
0
votes
2answers
38 views

Show Function is increasing

Suppose that $x > 0$. Given a function $f(x) = x(1 - \mathrm{e}^{-x})$, what is the easiest mathematical way to find out if this function is monotone? It is easy to plug numbers on $x$ and see ...
1
vote
2answers
38 views

Sequence formed by points of continuity

Let $f:\mathbb{R} \longrightarrow \mathbb{R}$ be a monotone increasing function. It is well known that $f$ has at most a countably infinite number of discontinuities. I would like to know if, given a ...
2
votes
2answers
44 views

$f:\Bbb{R} \rightarrow \Bbb{R}$ uniformly continuous function such that the sequence $(f(n))$ is increasing .

$f:\Bbb{R} \rightarrow \Bbb{R}$ uniformly continuous function such that the sequence $(f(n))$ is increasing but there is no interval $(a,b)$ with $a<b$ such that $\forall a<x<y<b$, $f(x)&...
0
votes
0answers
31 views

A continuous function is one to one only if it is strictly monotone

Proposition: $f:[a,b] \to \mathbb{R}$ is one to one only if $f$ is strictly monotone. Proof: Suppose, the statement is false. Then, for some $c, d \in (a,b)$, $f(d)=\sup f=M, f(c)= \inf f=m $ [...
0
votes
1answer
46 views

Give an example of function that satisfies this theorem? [closed]

Give an example of function that satisfies this theorem ? Theorem :The set of points of discontinuity of a monotonic function $f :\mathbb{R} \rightarrow \mathbb{R}$ is at most countable ...
1
vote
0answers
19 views

Sequence of nonnegative measurable functions which decreases to a function f

Let $(f_n)$ be a sequence of nonnegative measurable functions which decrease pointwise to a function $f$ on a measurable set $E$, and suppose that $\int_E f_k < +\infty$ for some $k$. Prove that $\...
1
vote
2answers
45 views

Proving difference between two functions is small when $x$ is small…

I'm really struggling to prove the following claim and I was wondering whether anyone could help me. Claim:$$ 0<|x| \leq 10^{-4} \implies \bigg{|} \frac{x}{e^x-1} - \frac{1}{\sum_{k=1}^3 \frac{x^...
0
votes
4answers
59 views

Proof on property on increasing function [closed]

I struggle on the proof of the following property : let $f$ be an increasing function, and suppose $L=\lim_\limits{x\to +\infty} f(x)$ exists Then $$L\geq f(x)$$ Samely for a decreasing function : ...
0
votes
1answer
32 views

Is this ratio of difference of these functions monotonic?

If $f(x)$ is monotonic decreasing and $g(x)$ is monotonic increasing then is $([f(x)]^n - [g(x)]^n)/(f(x)-g(x))$ monotonicslly decreasing always or under any specific conditions? All functions above ...
4
votes
2answers
108 views

Is $f(x) = \sum_{n\geq 1} \frac{\cos n x }{\sqrt{n}}$ monotonic on $(0,0.1)$?

Is the function $f$ defined by$$f(x) = \sum_{n\geq 1} \frac{\cos n x }{\sqrt{n}}$$ monotonic on the interval $(0,0.1)$? By Dirichlet's test, the series converges on this interval. Does it define a ...
1
vote
1answer
94 views

On monotonic quadratic least squares

Quadratic least squares can be used to fit a quadratic curve to $3$ or more points, such that the resulting curve is the quadratic curve that has the least squared distance of the data points to the ...
1
vote
1answer
115 views

Decreasing $f:\mathbb{R}\to\mathbb{R}$ tending to $0$ at $∞$ not convex beyond any point?

Given a function $f:\mathbb{R}\to\mathbb{R}$ differentiable and strictly decreasing such that $\displaystyle \lim_{x\to\infty}f(x)=0$, I am looking to find out whether or not there exists an $x_0$ ...
0
votes
1answer
53 views

$(1-x)^2$ function

I had a question about checking whether $f(x)=(1-x)^2$ is completely monotonic. My argument is that it is not, because: It is not strictly monotonic on $[0, \infty)$. It does not satisfy the ...
0
votes
1answer
36 views

Monotonicity of $ f'$

Let $f(x)=\sin x-x+\frac {x^3}{6}$ and $g(x)=\cos x-1+\frac {x^2}{2}$ for $x\in \Bbb R$.Then How to prove that $f(x)\ge 0$ for all $x\ge 0$? From given function it is clear that $f'=g$ and $g'(x)=x-\...
0
votes
0answers
28 views

A monotone function must have a tangent ray that does not cross with the function

Consider a monotonically increasing and differentiable function $y=f(x)$ that passes through the origin. $\gamma=\{(x,y)|y=f(x)\}$ is the graph of $f$. Claim: there exists a ray $R$ such that $R\cap ...
0
votes
1answer
29 views

Check if 1-x is completely Monotonic function

I had a question about how can I check whether $f(x)=1-x$ is completely monotonic. Could somebody provide a simple example based on this function.
0
votes
0answers
21 views

Looking for a functional form $f(x_1, x_2, x_3)$, increasing in $x_1, x_3 $ and $x_1-x_2$

I need to construct a simple function with three elements: $x_1, x_2$, and $x_3$. $x_1$ and $x_2$ are variables between $0$ and $1$. $x_3$ is positive and larger than $1$ I need $f(x_1, x_2, x_3)$ be ...
2
votes
2answers
38 views

Proving a function is monotone

Let $n\in \mathbb{N}$, $u_1,u_2,\ldots ,u_n>0$ and I want to prove that the function $$p(\alpha)=\frac{\sum_{i=1}^n u_i^\alpha}{\left( \prod_{i=1}^n u_i^\alpha \right)^{1/n}}$$ is monotone in ...
6
votes
2answers
208 views

Proving concavity of derivative

Let $f(x)$ be defined and continuous and derivable for $x>-1$, $f(0)=1$, $f’(0)=0$ and $$f''(x) = \frac {1+x}{1+f(x)}.$$ Prove that $f’(x)$ is concave up for all $x>-1$. My attempt: I ...
0
votes
1answer
22 views

When is a positive matrix a monotone operator?

Let $A$ be an $n \times n$ matrix with all elements positive, and $\lambda$ its largest eigenvalue, which is real, positive, and of multiplicity one. Define $M = (1/\lambda) \, A$. Let $x$ be a ...
2
votes
1answer
81 views

If $f:[0,t]\to[0,t]$ is continuous and increasing, why can we conclude $f(0)=0$ and $f(t)=t$?

Let $t\ge0$ and $f:[0,t]\to[0,t]$ be continuous and (strictly) increasing. Why can we conclude $f(0)=0$ and $f(t)=t$? I've tried the obvious thing: Let $\varepsilon>0$. Since $f$ is increasing and ...
0
votes
0answers
15 views

Existence of striclty monotone transformation

Assume we have a function class $F$ containing bivariate functions $f(x,y)\; (f: \mathcal{X} \times \mathcal{Y} \to \mathbb{R})\ $ that are continuously differentiable with respect to each argument. ...
0
votes
1answer
121 views

Estimation of fractional expression

We define $$\displaystyle f(x,y)=\frac{1}{x^{2y}-\frac{1}{4^y}}+\frac{1}{(1-x)^{2y}-\frac{1}{4^y}} \text{ for } (x,y) \in \left[0,\frac{1}{2}\right) \times \left(\frac{1}{2},1\right]$$. A study ...
0
votes
1answer
16 views

Boundedness of derivative of bounded, monotonous, continuously differentiable function

Let $f\in C^1(\mathbb{R})$ be bounded and monotonous. What else do we need from $f$ for its derivative $f'$ to be bounded, too?
4
votes
2answers
55 views

When can we use derivative test to identify injective function?

I have doubt regarding first derivative test for identifying whether a function is injective or not: For example: $$f(x)=\ln x$$ has domain $(0, \infty)$. Now $$f'(x)=\frac{1}{x} \gt 0$$ hence $f(...
0
votes
1answer
32 views

One-sided limits of a monotonic function

Let $f: \mathbb{R} \to \mathbb{R}$ be an increasing function. I am trying to prove that for any 2 distinct points of discontinuity $a,b$ of this function, if $a<b$, then $f(a^+)<f(b^-)$. I have ...
0
votes
3answers
39 views

How to show that one not monotonous f doesn't have fixpoints?

I have a question about fixed points If I have one function $f$ (that is not monotonous!) I would like to demostrate that this function hasn't fixed points. I need to find a funciton $f$ for which ...
2
votes
1answer
57 views

If $\mathbb E[X]$ is increasing in $\theta$, is $\mathbb E[X^2]$?

Consider a discrete random variable $X\in[0,1]$ with p.m.f. $f_X(x)$ parametrized by $\theta$. Assume its expected value $\mathbb E[X]$ is increasing in $\theta$. Is $\mathbb E[X^2]$ increasing in $\...
-2
votes
2answers
30 views

If $f(t)$ is continuous for $t$ $\in [0,1]$, $f' > 0$ and $f''(t) > 0$ for $t \in (0,1)$, do we have that $f'(t)$ is strictly increasing on $[0,1]$.

If $f(t)$ is continuous for $t$ $\in [0,1]$, $f' > 0$ and $f''(t) > 0$ for $t \in (0,1)$, do we have that $f'(t)$ is strictly increasing on $[0,1]$? Here is what i think: Since $f(t)$ is ...
0
votes
2answers
22 views

How to show this function is increasing both intuitively and using formula

In a book, it is claimed that the following function is obviously increasing as $x$ gets larger. I tried a few numerical examples for $x$, and it appears to be so. However, it is not clear to me how ...
0
votes
0answers
34 views

Does $f(x_n^+)=\inf\limits_{x_n<t< b}f(t)=\sum\limits_{x_i\leq x_n}c_i$ hold? Remark 4.31 in “Principles of Mathematical Analysis” by Walter Rudin.

Let $E$ be a countable subset of $(a, b)$. Let $\phi$ be a bijection from $\mathbb{N}$ to $E$. Let $\{x_n\}$ be a sequence such that $x_n := \phi(n)$. Let $\{c_n\}$ be a sequence of positive numbers ...
2
votes
2answers
62 views

Is ‎$‎f‎$‎ ‎monotone ‎when ‎$‎f‎$ ‎is ‎concave?‎

‎Let ‎$‎f:[1, +‎\infty‎)‎‎‎\rightarrow‎‎\mathbb{R}$ ‎be a‎ ‎concave ‎function. Suppose‎ $‎F:[1, +‎\infty‎)‎‎\rightarrow‎‎\mathbb{R}‎$ is a primitive function of ‎$‎f‎$‎. My ‎questions ‎are‎:‎ ‎‎ ‎(a) ...
1
vote
1answer
28 views

Construct a new function

I want to construct a function $f(x,y)$ ($x$ and $y$ are positive integers) which satisfies the following properties: 1) For each fixed $y$, (a) $f(x,y)<f(x+1,y)$ when $x<y$ (b) $f(x,y)>f(...
1
vote
1answer
109 views

Are there any equation that could produce a monotonic smooth step function by parameter

I want to write a mathematic formular that, given any number of monotonic arbitrary point, it will produce a monotonic smooth step function Such as a figure below, I give it 2 point (the intersect ...