# 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.

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### $(1-x)^2$ 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) ...
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### monotone function without derivative test

How i can prove this function is not a monotone function without the derivative test? $$f(x)=-\frac{1}{x^3}$$ thanks in advance
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### Monotonicity of $a_n=1+\frac{(-1)^n}{n}$

I'm trying to study the monotonicity of $a_n=1+\frac{(-1)^n}{n}$, but what I'm getting isn't correct: I just assume that $a_n$ is monotonically increasing, and if it isn't, I'll get something absurd: ...
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### $x_n=1+\frac{1}{\sqrt2}+…+\frac{1}{\sqrt n}-2\sqrt n$

$x_n=1+\frac{1}{\sqrt2}+...+\frac{1}{\sqrt n}-2\sqrt n$ From here Investigating the convergence of a series using the comparison limit test, Part II I can see that the sequence converges. I was ...
What does the notation $f(x^{+})$ and $f(x_+)$ mean? The context is the following I have a proposition concerning monotonic increasing functions, so $f$ is nondecreasing, also $x\in(a,b) =I$ where $f$...