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.
1,314 questions
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Showing the strict mononticy of the function $(a^\frac{1}{x}+b^\frac{1}{x})^x$ [closed]
Let $a,b\in \mathbb{R}$ with $a,b>0$. Define the function $f\colon [1,\infty)\rightarrow \mathbb{R}$ by
$$
f(x)=(a^\frac{1}{x}+b^\frac{1}{x})^x
$$
This function seems to be strictly growing, but I'...
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Counterexample to "the square of a non-monotone function is non-monotone"
The question: “If a function is not monotone on $(a, b)$, then its square cannot be monotone on $(a, b)$.” We are to provide a counterexample to this statement.
On initial attempts I was able to forge ...
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Let $f: \mathbb{R} \to \mathbb{R}$ (differentiable, one -one)such that its limit at infinity exists, then $\displaystyle \lim_{x \to \infty}f'(x)$? [duplicate]
Let $f:\mathbb{R} \to \mathbb{R}$ be a differentiable and one-one function such that
$$\lim_{x \to \infty} f(x)$$
exists and is finite.
What can we say about
$$\lim_{x \to \infty} f'(x)?$$
My ...
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$x^2$ is absolutely monotonic but does not satisfy Bernstein's theorem (unless...)
I've been working for a while on something related to absolutely monotonic functions, and I've come to this realization and it feels like a significant mistake in the literature that I cannot believe ...
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If a sequence is monotonic for all $n$ sufficiently large and bounded, then is it convergent?
Edited
I've recently been trying to show that in $\mathbb{R}$.
If the sequence $(x_n)$ is monotonic for sufficiently large $n$ and it is also bounded, then $(x_n)$ is convergent.
I'll present my ...
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Monotonicity of the surface area of a twisted square as a function of rotation angle
Let $ABCD$ be the unit square.
$A(0,0,0),\; B(1,0,0),\; C(1,1,0),\; D(0,1,0)$
For $k\in[0,1]$, define
$$
P(k)=(k,0,0),\qquad Q_0(k)=(k,1,0).
$$
Now rotate the top edge $CD$ by an angle $\theta$ around ...
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If $g'(x)=f(g(x))$ for decreasing derivable $f$ on $(0,+\infty)$ and positive derivable $g$ on $\mathbb{R}$, $g'(x)$ has a zero, is $g(x)$ constant?
Let $f$ be a decreasing derivable function on $(0,+\infty)$, and $g$ be a positive derivable function on $\mathbb{R}$. If $g'(x)=f(g(x))$, and $g'(x)$ has a zero, is $g$ a constant function?
I set $...
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A condition on all derivatives at $x=0$ implies $f\geq0$?
Let $f:\mathbb R\to\mathbb R$ real analytic function on $\mathbb R$. Suppose that its Taylor series around $x=0$ is
$$ S(x) = \sum_{k=0}^\infty (-1)^k\,a_k\,x^k$$
where $a_k\geq0$ for all $k$. In ...
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A function of two variables separately continuous and monotone is (jointly) continuous
In the paper
Ciesielski and Miller. A continuous tale on continuous and separately continuous functions. Real Analysis Exchange Vol. 41(1), 2016, pp. 1-36.
the following theorem appears:
Theorem 11. (...
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Monotonicity knowing all derivatives at $x=0$
Let $f:\mathbb R \to \mathbb R$ real analytic function.
Suppose that $f^{(k)}(0)\geq0$ for all $k=0,1,2,\dots$ (where $f^{(0)}$ denotes the functions itself and $f^{(k)}$ the $k^{th} $ order ...
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Show a function defined by series of functions is strictly increasing
I met an interesting function defined by series of functions, which is given as follows
\begin{equation*}
f(n,s)=\frac{s}{n(n+1)}-\sum_{k=0}^{+\infty}\frac{2s}{(sk+n)(sk+n+1)(sk+n+2)}.
\end{equation*}
...
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Show the monotonicity of a function involving the difference of the digamma function
Recently, I met an interesting function when studying higher-order moments of a class of distributions, which is given as
\begin{equation*}
g(n,s) = \psi\left(\frac{n+2}{s}\right)+\psi\left(\...
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Are all monotone Boolean functions weighted threshold functions?
$\DeclareMathOperator{\th}{th}$
A Boolean function $f:\{0,1\}^n \to \{0,1\}$ is monotone if for all $a_1,\dots,a_n,a'_1,\dots,a'_n$, if $a_i \leq a'_i$ for all $1 \leq i \leq n$ then $f(a_1,\dots,a_n) ...
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Monotonicity of Hardy Computation breaks when switching addends?
I am working with ordinal arithmetic at the moment, more precisely with ordinals below epsilon zero. Further I am considering the Hardy computation as a ordinal indexed hierarchy of functions as ...
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143
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Number of solutions to the diophantine equation $1/a+1/b+1/c=1/p(k)$, where $p(k) = k$-th prime
Playing around with diophatine equations of the harmonic mean type I started with
$$\frac{1}{a}+\frac{1}{b}=\frac{1}{p}\tag{1}$$
where, for simplicity, $p$ is a prime number and asked for the number ...
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Function with incompatible properties?
For didactical / illustrative pourposes I’m searching a real valued function with the following properties:
$\mathcal{C}^\infty$ over all $\mathbb{R}$ or better analytic over the entire complex plane....
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Under what conditions can a continuous multivariate function be represented as a function of a sum?
Suppose that
$$
f(x_1, \ldots, x_N) \rightarrow \mathbb{R}, x_i\in\mathbb{R},
$$
and the following conditions hold:
$f$ is analytic.
The arguments to $f$ are exchangeable in the sense that any ...
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Is the zero set of $f’$ is relative closed when $f$ is increasing, uniformly continuous and differentiate everywhere in an open interval?
In my analysis textbook, there is a theorem stating: Let $f$ be a uniformly continuous and differentiable function defined on $(a, b)$. Then the following statements are equivalent:
"$f$ is ...
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73
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Is it really true that the (left-) inverse of a strictly increasing function is strictly increasing?
There is a question here on the site that claims that if a function $f:X\rightarrow Y$ is strictly increasing, then also its left inverse $f^{-1}$ is strictly increasing.
Inverse of Strictly Monotone ...
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If $J$ is a bounded interval and g is non-decreasing and continuous then $\sup(g(g^{-1}(J))) = \sup J$.
I have been going through Tao's Analysis 1 textbook and I am now stuck on a particularly difficult question at the very end that I've managed to reduce the proposition down to this:
If $J$ is a ...
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1
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Identifying a monotone transformation of a linear function
Given a function $f: \mathbb{R}_+\to \mathbb{R}_+$, is there a way to decide whether there exist function $g,h$ such that:
$f(x) \equiv g(h(x))$;
$g$ is a monotonically-increasing function;
$h$ is a ...
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Proving the uniqueness of a fixed point on $(0,1]^N$
I'll preface this by saying I am not very experienced with formal proofs, so that's mainly why I'm asking for help here. I found several other threads on this topic here on Math SE, but most of them ...
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Approximation scheme or bounds for a completely monotonic sequence, for example $U_n=(1+n)^{-2}$
Definitions: let a sequence $U_n$ decrease to zero. Further suppose it to be completely monotonic, i.e.:
$\displaystyle (-1)^{k}\Delta ^{k}U_{n}\geq 0, \quad n,k=0,1,2,\ldots$
where $\Delta^kU_n=\...
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views
Is $H(e^{-2s})/H(e^{-s})$ completely monotone, where $H(\cdot)$ is the binary entropy?
While tacking the related problem, I came up with the following question: Let $H(x) = -x \log x - (1-x)\log(1-x)$ be the binary entropy, and consider the function
$$ F(s) = \frac{H(e^{-2s})}{H(e^{-s})}...
- 185k
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Delete smallest number of points to make a sequence monotonic
This is actually motivated by a real problem I am having at work. Suppose we have a sequence $x_1,...,x_n$ and some collection of points (which we call spikes) $k_1,...k_m \in \{1,...,n\}$ which, if ...
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Prove that the sequence $x_{n+1}=x_n-\frac{b-x_n}{f(b)-f(x_n)}f(x_n)$ is strictly increasing
Let $f:[a,b]\to\mathbb R, f \in C^2[a,b]$. Suppose $f$ is convex in $[a,b]$ and $f(a)<0, f(b)>0$. With these hypotheses, the method of false position produces the following sequence: $$x_{n+1}=...
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Monotonic interpolation of three points
I am trying to implement a smooth $C^2$ spline. The paper I am trying to implement seems to require a reparametrization of a Bezier curve. I am looking for the reparametrization function $\phi(\theta)$...
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Does directional derivatives positive in all directions imply monotonicity?
Consider the projection $p(x,y)=\frac{x}{1-y}$ from the domain (unit circle minus the north pole $(0,1)$) to $\mathbb{R}$. Since it is projection, it is injective.
My question: Can we conclude the ...
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Is there a strictly increasing function from any order theoretic tree of countable height to the reals such that the supremum over any branch is 1?
Let $(T,\leq)$ be a tree with $ht(T)=\alpha<\omega_1$. Is there a strictly increasing function $\psi:T\to[0,1]$ (i.e. $\forall a,b\in T:a<b\Rightarrow \psi(a)<\psi(b)$) such that for every ...
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96
views
Which increases faster? $ -\ln(y(t))$ or $\gamma t$?
I have a function $y(t) > 0$ for $t \ge 0$, such that $\lim_{t \rightarrow \infty} y(t) = 0$. Where $C < 0, \:\: \gamma > 0$ be some constants. How can I show that there is some $T > 0$ ...
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Why do lateral limits exist for this monotone function?
I'm trying to understand a solution to the following problem from Elon Lima's Real Analysis.
Prove that the set of discontinuity points of a monotone function is countable.
The proof goes like this. ...
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Do antitonic, subidempotent functions have to be constant under a total order on $X$ ($|X| > 1$)?
Assuming $\leq$ is a total order, in other words a reflexive, transitive, anti-symmetric, connected (every pair is comparable) binary relation, on a set $X$ of cardinality $\geq2,$ does the condition $...
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How to prove pair of recurrence relations are monotonic and converge to $\pi$ [duplicate]
Let $a_{0}=2\sqrt{3}$ and $b_{0}=3$, and define two sequences recursively by:
$$a_{n}=\dfrac{2a_{n-1}\cdot b_{n-1}}{a_{n-1} + b_{n-1}} \; \; \text{and} \; \; b_{n}=\sqrt{a_{n}\cdot b_{n-1}}$$
Prove ...
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Why does the minimum of monotonic square root of a function is the same as the mimimum of the function under root. [closed]
All,
To deduce the regression coefficient $$c=(A^T A)^{-1} A^T B $$, we assume that the minimum of the square root error $\sqrt{\sum({y_i - y_{hat})}^2}$, reduces to finding the minimum of the ...
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33
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Closed curve reparametrization
Let say, I have a smooth periodic trajectory $\gamma\left(t\right)$ in phase space $\mathbb{R}^{n},n>2$. I want to reparametrize the trajectory using new $\theta$. To this end, I find projection of ...
- 220
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1
answer
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views
Finding a Rational Number in an Interval with Strict Monotonicity Requirement [closed]
I tried using floor functions to find a rational number in the given interval with strict monotonicity, but didn't work.
Given a < b with a, b ∈ ℝ (or equivalently, given a and d where d > 0 ...
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votes
2
answers
218
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Monotonicity of a specific function
Consider a positive, increasing, continuous, differentiable function $F(\cdot)$ such that $F^{n-1}(\cdot)$ is convex (the superscript is an exponent). For any two values, $x>y$, and natural $n>1$...
- 101
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30
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Equivalence of definitions of monotonicity
Suppose I have a function $F:\mathbb{R}^n\to \mathbb{R}^n$. I have seen two definitions of monotonicity of $F$: one being with respect to a partial order $\succ$, which states that $F$ is monotone if $...
- 1,337
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vote
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80
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Proof of monotonicity and concavity for the difference of binary entropies with different arguments
Let $$h(x) = -x\log(x)-(1-x)\log(1-x)$$ is a binary entropy function and $$f(x) = \frac{1-\sqrt{1-4x(1-x)b}}{2},$$ for some $0\leq b \leq 1$ ($0\leq x \leq 1$ too). I need to prove two statements:
$g(...
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56
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$Cov(f(x+y), x )<0$ for strictly decreasing function $f$ and $Cov(x, y) \geq 0$
Let $x, y$ be well-behaved continuous random variables, $f(\cdot)$ be a differentiable real-valued function with $f'<0$. Assume $Cov(x, y) \geq 0$. My intuition says
$$Cov(f(x+y), x)<0$$
Is that ...
- 21
2
votes
1
answer
68
views
Monotonicity including hyperbolic function equation with parameter
A hyperbolic function $\varphi(x)$ is defined as
$$\varphi(x)=\frac{b-\delta}{b-a}\text{sinh}(\frac{a}{x})+\frac{\delta-a}{b-a}\text{sinh}(\frac{b}{x})-\text{sinh}(\frac{Q}{x})$$
under the condition $...
- 33
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votes
1
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107
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Arithmetic expression for lexicographically monotonic function over counting numbers and into the reals?
Motivation
I have two motivations for this question. The primary one is something I want in order to meet the requirements of a software tool I am developing. The ancillary motivation is to have a ...
- 1,946
0
votes
1
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48
views
Arithmetic expression for lexicographically monotonic function for natural numbers?
I am looking for an arithmetic expression for a map $\phi:\mathbb{N}_0^n \mapsto \mathbb{N}_0$ such that
$$a,b\in\mathbb{N}_0^n$$ and $$a <_{\text{lex}} b$$ then $$\phi(a) < \phi (b).$$
The ...
- 1,946
0
votes
0
answers
43
views
Formalizing intuition for ratio fo consecutive terms of sequences
I am curious about the ratio of subsequent terms of sequences but am having trouble formalizing intuition for this. Specifically, consider two strictly monotone sequences $(a_{n})$ and $(b_{n})$ such ...
5
votes
1
answer
218
views
Proving convergence of a sequence via the Monotone Convergence Theorem
Investigate the convergence (and determine the limit, if convergent) of the sequence below. Justify all steps carefully.
$$a_{n+1}=\sqrt{2+a_n}, n\geq2$$ $$a_1=1$$
Note: I am interested in alternate ...
- 2,674
3
votes
1
answer
204
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Find the monotone sequences in $\frac{m^2+n^2+2a_n\cdot\, a_m}{ {a_m}^2 + {a_n}^2 + 2mn} \in \mathbb{N}$
The Problem
Determine the monotone sequences of nonzero natural numbers $(a_n)_{n\geq 1}$ with the property that $$\frac{m^2+n^2+2a_n\cdot\, a_m}{ {a_m}^2 + {a_n}^2 + 2mn} \in \mathbb{N}$$ for any $m,...
user1353144
3
votes
1
answer
70
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Continuity of a function defined implicitly via level sets
Let $X$ be a topological space and let $F:X\times[0,1]\to[0,1]$ be continuous. Suppose that for each fixed $x\in X$, the map $f_x:[0,1]\to[0,1]$ given by $f_x(t):=F(x,t)$ is monotone and assumes the ...
- 1,371
2
votes
1
answer
134
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Does such a factorization exist?
Let $\theta_2,\theta_1$ be two real positive parameters with $\theta_2>\theta_1$ and consider the function \begin{align}F_{\theta_1,\,\theta_2}(x)=x^{\theta_2-\theta_1}\exp\left(-x^{\theta_2}+x^{\...
- 1,954
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0
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31
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Condition for Monotonic Phase Increase in $ z^2 + bz + c$ over the Unit Circle
I am studying a quadratic polynomial $P(z) = z^2 + bz + c$ , where $b$ and $c$ are real constants. I want to determine conditions on $b$ and $c$ such that
The phase of $P(z)$ is monotonically ...
3
votes
1
answer
90
views
Limits and monotonicity to find function
Given functions $f, g: \mathbb R\rightarrow\mathbb R$, $f(x)$ is monotonic and $\lim\limits_{x \rightarrow a^+}f(x)=g(a)$ $\forall a \in \mathbb R $. How can I show that if $g$ is continuous at some $...
