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.

Filter by
Sorted by
Tagged with
2 votes
0 answers
20 views

Knowing $f(t^\ast)\ge 0$ (and some other information), can we show that $f(t)\ge 0$ at $t<t^\ast$?

I asked a similar question before and had to make several changes so before anyone spends time on answering it, I decided to clarify here. We have constants $\alpha_1,\alpha_2,r_1,r_2,c>0$ and we ...
user avatar
0 votes
0 answers
24 views

Difference of monotonic functions

Consider the strongly monotonic function $w:\mathbb{R}\rightarrow [a\;b]$, with $a,b\in\mathbb R$, that is, there exist a scalar $\eta(x,h)$ that verifies $$ \frac{w(x+h)-w(x)}{h}\geq \eta(x,h)>0, \...
user avatar
  • 29
1 vote
1 answer
46 views

How to take the derivative with respect to a function without a clear substitution?

In Statistical Inference, the Karlin-Rubin Theorem requires that a given statistical model has a Monotone Likelihood Ratio with respect to a sufficient statistic $T(X)$. In order for the model to ...
user avatar
  • 991
3 votes
2 answers
40 views

Why does the monotone convergence fail here

I am looking at the example $f_{n}(x) = n \chi_{(0, \frac{1}{n}]}$. This converges to $0$ pointwise and graphing it out we can see that its a series of rectangles of area 1 but with growing height. I ...
user avatar
0 votes
1 answer
33 views

If $\frac d{dx} f(x,y)>0$, can I claim that $f(x,y)$ is increasing with respect to $x$?

I have an implicit equation $f(x,y)=0$; computing the derivatives, I see that $\frac d{dx} f(x,y)>0$ while $\frac d{dy} f(x,y)$ maybe positive, or negative. Question. Is this data sufficient to ...
user avatar
  • 155
5 votes
1 answer
137 views

Proving $f_n \rightrightarrows f$, provided that $f_n \rightarrow f$, each $f_n$ is increasing, and $f$ is continuous.

Clarification: this is not Dini's theorem. The title is highly succinct but here is the task in full detail: Let $[a,b]\in \mathbb{R}$ where $a<b$ are reals. Each $f_n: [a,b] \to \mathbb{R}$ is an ...
user avatar
  • 394
0 votes
0 answers
30 views

Show that every solution of the differential equation is monotonic

I just started learning ODE and found this problem. Can anyone help me prove it. Let $f : \mathbb{R} → \mathbb{R}$ be continuous. Show that every solution of the differential equation $x' = f(x)$ is ...
user avatar
0 votes
0 answers
21 views

Generating a polynomial which goes through $n$ points and is monotonically non-decreasing

Suppose we have $x_1<x_2<x_3<\cdots<x_n$ and $y_1<y_2<y_3<\cdots<y_n$. I was wondering if it would be possible to find a polynomial $P$ such that $P(x_i)=y_i$ for all $1\leq i\...
user avatar
  • 2,615
2 votes
0 answers
92 views

Category of partitions of an interval

Revisiting the definition of Riemann integral, Carla noticed that we can define partitions of an interval categorically: given a (nondegenerate) interval $[a,b]$ seen as an ordered set with top and ...
user avatar
  • 41
0 votes
0 answers
16 views

extremum points of function

I was wondering if I have a function F[x,t], (in my case a polynomial), and I find the extremum points, which are fractions, if the denominator of the extremum points vanishes, does this ensure ...
user avatar
  • 1
6 votes
1 answer
133 views

monotone functions agreeing with Holder functions on a large set

Let $\alpha \in (0,1)$, $f:[0,1]\rightarrow \mathbb{R}$ be a continuous monotone function and $\varepsilon>0$. Does there exist a function $\phi_{\varepsilon} \in \mathcal{C}^{\alpha}$ such that $\...
user avatar
  • 41
0 votes
0 answers
33 views

Why can we assume WLOG $\alpha$ is increasing?

I have a question regarding the proof of Theorem 9.8 from Mathematical Analysis by Tom Apostol below: Theorem 9.8: Let $\alpha$ be of a bounded variation on $[a,b]$. Assume that each term of the ...
user avatar
1 vote
0 answers
21 views

common fixed point of commuting functions of unit interval

Let $f,g:[0,1]\to[0,1]$ commute ($f\circ g=g\circ f$). Suppose $f$ is weakly increasing and $g$ is continuous. Is it necessarily true that there exists $x\in[0,1]$ such that $x=f(x)=g(x)$ ? Motivation....
user avatar
  • 115
0 votes
0 answers
20 views

Generating Monotone Boolean Function

I recently went thru an article over Generating Monotone Boolean Function (at https://www.mathpages.com/home/kmath094/kmath094.htm ) but couldn't understand the concept of using 2 monotone functions ...
user avatar
  • 1
2 votes
1 answer
58 views

Is this function monotonically increasing as $x_2$ increases?

Suppose I have a differentiable and continuous function $f(x)>0$, the monotonicity of $f(x)$ is unknown. Assume that $x_1< x_2< x_3 \in \mathcal{S}$, $\mathcal{S}$ is the domain of $f(x)$. ...
user avatar
  • 161
2 votes
2 answers
71 views

Finding the intervals of increase and decrease of $\frac{x^4 - x^3 -8}{x^2 - x - 6}$

How can I find the intervals of increase and decrease of $\frac{x^4 - x^3 -8}{x^2 - x - 6}$? I tried to find the derivative by the quotient rule to obtain the critical points but the formula was ...
user avatar
  • 1,191
2 votes
1 answer
27 views

A mapping of two sequences with no overlaps and partial assignments

I want to characterize a correspondence mapping of two sequences $\psi : A \rightarrow B$ for an article that I am writing. I need help describing the function class. I think this is an injective, ...
user avatar
0 votes
1 answer
27 views

How to show that the maximum value of a given continuous function over epsilon ball is continuous

I'd like to show that the following statement: Suppose that a continuous function $f: \mathbb{R}^n \to \mathbb{R}$ is given. Let $B_{\epsilon}$ be a ball at the origin with the radius of $\epsilon \...
user avatar
0 votes
0 answers
13 views

How to prove the hazard ratio function of two beta distributions is monotone decreasing?

Suppose there're two random variables $X \sim Beta(a, b_1)$ and $Y \sim Beta(a, b_2)$ with $a \in (0, 1)$ and $b_1 < b_2$. Then the hazard ratio function is $$\theta(u) = \lambda_X(u) / \lambda_Y(u)...
user avatar
  • 161
2 votes
1 answer
80 views

How many monotonic functions on sets of naturals are there?

A function $f\colon P(\mathbb{N}) \to P(\mathbb{N})$ is monotonic iff $x \subseteq y \implies f(x) \subseteq f(y)$. What is the cardinality of the set $F$ of all such functions? What I have tried: I ...
user avatar
2 votes
3 answers
88 views

Prove that if $\lim _{n\rightarrow \infty }a_{n}=1$ then $\lim _{n\rightarrow \infty }a_{1}+a_{2}+\ldots +a_{n}=\infty $

Would appreciate some help with proving the following statement: Let $a_{n}$ be a sequence, and $\lim _{n\rightarrow \infty }a_{n}=1 $. Prove that $\lim _{n\rightarrow \infty }a_{1}+a_{2}+\ldots +a_{n}...
user avatar
  • 67
0 votes
0 answers
30 views

Existence of strictly increasing functions with added property

Given a weakly increasing function, I would like to prove the existence of strictly increasing functions (or even construct one) that lie above the weakly increasing function and agree with it at ...
user avatar
  • 1
2 votes
2 answers
68 views

Riemann integrable function on [a, b] which is not monotonic [a, b]. [duplicate]

I need to find a Riemann integrable function, which is not monotonic on a closed interval. But I couldn't find one. I checked some continuous and discontinuous functions but still couldn't find a ...
user avatar
  • 217
0 votes
2 answers
29 views

How do I determine the monotonicity of $y=2^{\frac{1}{x-a}}$?

I've worked out the derivative and the critical point: $f'(x)=-\frac{2^{\frac{1}{x-a}}\ln 2}{(x-a)^2}$ and $x-a=0\iff x=a$, but I don't know where to go from here.
user avatar
  • 329
1 vote
1 answer
23 views

How can I show monotonicity of this function defined on the space of random variables $RV(\Omega)$?

I have the function $f_{\lambda}:RV(\Omega)\rightarrow \mathbb{R}$ defined on the space $RV(\Omega)$ supported over some scenario set $\Omega$: $f_{\lambda}:=\frac{1}{\lambda}\log(\mathbb{E}[e^{-\...
user avatar
0 votes
0 answers
29 views

$f:[a,b]\to\mathbb{R}$ is a monotonic increasing function that takes on every value $c\in[f(a),f(b)]$. Show that $f$ is continuous

I am preparing for my exam and need someone who could check, if my solution for the following task is correct: Let $f:[a,b]\to\mathbb{R}$ be a monotonic increasing function that takes on every value $...
user avatar
1 vote
0 answers
107 views

Is $f(x) = x^{x^x}$ monotonic?

How can we prove that $f_3 \colon (0, +\infty) \to \mathbb{R}$, $f_3(x) = x^{x^x}$ is monotonic? I tried taking the derivative — didn't work. Can we extend the proof to $f_{2k+1}$ for any $k$ (this ...
user avatar
  • 141
0 votes
1 answer
27 views

Check the monotonicity of an inductive sequence

Given $$a_{n+1} = \sqrt{\dfrac{ab^2+a_n^2}{a+1}}\,\,\forall n$$ Where $a>0, 0< a_1< b, a=a_1$ I obtained the expression for terms $a_2^2, a_3^2, a_4^2,...,a_n^2$ and do the summation of $a_i^...
user avatar
0 votes
0 answers
17 views

Function derivative and monotonicity

Why do we say f'(x) >= 0 for all x in the interior of I exactly if f(x) is increasing on all of I; Instead of f'(x) >= 0 for all x in the interior of I exactly if f(x) is nondecreasing on all ...
user avatar
  • 101
0 votes
1 answer
33 views

Show/"sketch" which area f is strictly positive.

I have the following question: Let $c>0$ and consider the following function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ given by: $$f(x,y)=\begin{cases} c y e^{-x} & \text{0 < x < $\infty$...
user avatar
  • 1
1 vote
0 answers
32 views

Prove that if a function is 2-monotone on $(a, c]$ and $[c, b)$, then it is 2-monotone on $(a, b)$

I am studying matrix monotone and I meet some difficulties when trying to prove the following statement: Let $f$ be a differentiable function on the interval $(a,b)$ such that $f$ is 2-monotone on $(a,...
user avatar
  • 11
0 votes
0 answers
26 views

Is there any upper-bound function such as $\frac{k}{x}$ for monotonically decreasing functions?

Could you please tell me if the following statement is true? "Any monotonically decreasing function can be upper-bounded by function $f(x)=\frac{K}{x}$, i.e, if $g(x)$ is a monotonically decaying ...
user avatar
  • 11
0 votes
0 answers
26 views

How does monotonicity guarantee that the Fourier series is exactly convergent to the given function (Parseval's identity)?

In Piskunov's calculus, Bessel inequality is proved as follows: They start with a periodic function $f(x)$ having period $2\pi$ whose Fourier coefficients are $(a_r,b_r)$. Suppose $$T_n:=\{ s_n(x) = \...
user avatar
0 votes
1 answer
54 views

Monotonically decreasing or increasing sequence

Let ${x_n}$ be a sequence of real numbers. consider the set $$P=\{n\in \mathbb{N}:x_n>x_m \forall m \in \mathbb{N}\hspace{4 pt} with \hspace{4 pt}m>n \}$$ Then which of the following is/are ...
user avatar
  • 320
0 votes
0 answers
36 views

Does this function is monotonically incresing about a?

Suppose that $\boldsymbol{t}\sim \mathcal{N}(\boldsymbol{u};\boldsymbol{0},\boldsymbol{M})=f_{\boldsymbol{t}}(\boldsymbol{u})$, where $\boldsymbol{t}$ is a $N$-dimensional gaussian random vector, and \...
user avatar
  • 161
0 votes
0 answers
26 views

How to solve for coefficients of monotonically increasing polynomial that must pass through two states?

Preface: I posted a similar question yesterday but wanted to change it slightly. I have a problem I can solve, but I want to modify it slightly and I'm unsure of the best way to solve the new problem. ...
user avatar
  • 11
1 vote
0 answers
70 views

How to solve for the coefficients of a polynomial such that the function is monotonically increasing?

Currently, I am solving for the coefficients of a quintic polynomial that passes through two states: $f(x) = c_0 + c_1x + c_2x^2 + c_3x^3 + c_4x^4 + c_5x^5$ The states define $f(x)$, $f'(x)$, and $f''(...
user avatar
  • 11
0 votes
0 answers
39 views

Quantile of Centered Binomial

Let $X$ be a Binomial distribution with $n$ trials and success probability $p$ in $(0,1)$. It is clear that the quantiles of $X$ are an increasing function of $p$. Let $Y$ be a centered Binomial, i.e. ...
user avatar
0 votes
0 answers
17 views

Integral decay implying pointwise decay

Let $\varphi : [1,\infty) \to [1,\infty)$ be a non-decreasing function such that $$ \int_1^{\infty} \frac{\mathrm{d}t}{\varphi(t)} = + \infty. $$ Then does there exist $p>1$ and $C>0$ such that $...
user avatar
  • 6,038
0 votes
0 answers
16 views

Show that an equation with $log^2(\cdot)$ is homothetic?

I need to show that the following function $v$ is homothetic (i.e., that there exists a strictly increasing function $g: \mathbb{R} \rightarrow \mathbb{R}$ and a homogenous function $u: \mathbb{R}^n \...
user avatar
  • 23
1 vote
1 answer
35 views

How to prove monotonicity of $\exp(x)-\sin(\exp(-x))$?

I want to prove that $x< y \implies f(x) < f(y) \forall x,y \in [0,\infty)$ for the function $f(x)=\exp(x)-\sin(\exp(-x))$ without using derivatives. I know that $e^x$ is increasing but $\sin(e^{...
user avatar
  • 13
0 votes
0 answers
16 views

Compute limit of the integral of $n\log(1+\frac{f^3}{n^3})$ w.r.t the Lebesgue measure.

Here is the setup: Let f be a non-negative measurable function on $\mathbb{R}$ where $\int_\mathbb{R}f\text{d}m_1=2021$. Show that the limit exists and compute $\lim_{n\rightarrow \infty}\int n\log(1+\...
user avatar
1 vote
0 answers
58 views

Integral of (generalized) inverse functions with a twist

Consider a weakly increasing function $f : [0,1] \rightarrow [0,1]$ and define its generalized inverse (since an inverse does not necessariliy exist unless it increases strictly) as $$ f^{-1}(t):= \...
user avatar
  • 105
-1 votes
1 answer
82 views

Proving that given a bounded function $f:\mathbb{R}\to\mathbb{R}$, there exists a strictly increasing sequence $(x_n)$ for which $(f(x_n))$ converges [closed]

I don't know where to start with proving it. Thanks for the help. Let $f:\mathbb{R}\to\mathbb{R}$ be a bounded function. Prove that there exists a strictly increasing sequence $(x_n)$ such that the ...
user avatar
3 votes
1 answer
71 views

Show the existence of a sorting function

Let $(X,\leq)$ be a totally ordered set. A sort for $f\in X^n$ is an element $g\in X^n$ satisfying (i) $g$ is nondecreasing. (ii) $ g=f\circ \sigma$ for some permutation $\sigma:\{1,\dots,n\}\to \{1,\...
user avatar
  • 3,433
6 votes
3 answers
147 views

Proving that a function $f:[0,\infty)\to[0,\infty)$ satisfying these conditions is necessarily non-decreasing

I have a function $f: [0, \infty) \to [0, \infty)$ which is smooth. I also have that $f(0) = 0$ $f'(0) > 0$ $f''(x) \leq 0$, for all $x \in [0, \infty)$ It intuitively makes sense then that $f$ ...
user avatar
0 votes
2 answers
35 views

Asymptotically monotonic sequence

Let $(u_n)_{n \geq 1}$ be a sequence of real numbers such that $$u_n \sim c n$$ where $c>0$. Is it true that $(u_n)_{n \geq 1}$ is necessarily asymptotically increasing (i.e. that there exists an ...
user avatar
  • 617
0 votes
0 answers
17 views

If a function to be maximized is almost everywhere differentiable can we say the derivative=0 condition must hold almost everywhere?

We have a known, increasing function $p:[0,1]\rightarrow[0,1]$. We want to find conditions on functions $R_1(\cdot),R_0(\cdot)$, $R_1(\cdot)$ known to be strictly increasing and $R_0(\cdot)$ known to ...
user avatar
  • 1,067
1 vote
0 answers
31 views

Linear combination of "basis" to approximate any continuous monotone increasing functions

Define $$ \mathcal{A} = \{ f(x) | x \in [a,b],f(x) \text{ is continuous monotone increasing} \} $$ $\mathcal{A}$ is not a vector space since there is no additive inverse. Is there a subset $\mathcal{A}...
user avatar
  • 133
3 votes
2 answers
100 views

Logical conclusion from a function being neither increasing nor decreasing.

A function $g:[a,b]→\mathbb{R}$ is increasing if $∀x_1{<}x_2$ in $[a,b]$, $g(x_1)≤g(x_2).\quad$ Similar is the definition of a decreasing function. So, if a function $f:[a,b]→\mathbb{R}$ satisfies ...
user avatar
  • 163

1
2 3 4 5
22