Questions tagged [monotone-functions]

Question 1

Problem statement: Let $f:\mathbb{R} \to \mathbb{R}$ be defined by $$ f(x) = a_1^x + a_2^x + \dots + a_n^x, $$ where $n \in \mathbb{N}, \quad n \ge 3,$ and $a_1, a_2, \dots, a_n > 0,$ all of them ...

Question 2

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'...

Question 3

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 ...

Question 4

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 ...

Question 5

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 ...

Question 6

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 ...

Question 7

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 ...

Question 8

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 $...

Question 9

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 ...

Question 10

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. (...

Question 11

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 ...

Question 12

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*} ...

Question 13

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(\...

Question 14

$\DeclareMathOperator{\th}{th}$ 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) ...

Question 15

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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Questions tagged [monotone-functions]

Show that $f$ is strictly increasing on $\mathbb{R}_+.$

Showing the strict mononticy of the function $(a^\frac{1}{x}+b^\frac{1}{x})^x$ [closed]

Counterexample to "the square of a non-monotone function is non-monotone"

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]

$x^2$ is absolutely monotonic but does not satisfy Bernstein's theorem (unless...)

If a sequence is monotonic for all $n$ sufficiently large and bounded, then is it convergent?

Monotonicity of the surface area of a twisted square as a function of rotation angle

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?

A condition on all derivatives at $x=0$ implies $f\geq0$?

A function of two variables separately continuous and monotone is (jointly) continuous

Monotonicity knowing all derivatives at $x=0$

Show a function defined by series of functions is strictly increasing

Show the monotonicity of a function involving the difference of the digamma function

Are all monotone Boolean functions weighted threshold functions?

Monotonicity of Hardy Computation breaks when switching addends?

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