How $\limsup_{n\to\infty}\left[\frac{a_n}{n^\varepsilon}\right]^{1/\log\log...
QuestionLet $(a_n) \subseteq \mathbb R_+$ be a sequence of non-negative numbers such that$$\limsup_{n \to \infty} \left[ \frac{a_n}{n^\varepsilon} \right]^{1/\log \log n} = e^{1+\varepsilon}$$for some...
View ArticleFind functions $f(x)$, so that $f(f(x)) = 4x$.
$f$ is a function defined from real numbers to real numbers. There are three things that we know:$f(x)$ is differentiable at all points,$f(0) = 0$,$f(f(x)) = 4x$.What are the possible values of...
View ArticleCan difference quotient sets be null?
Recently, there are several questions regarding what the difference quotient set - $\{\frac{f(x) - f(y)}{x - y}: x \neq y\}$ of a non-linear (or, more precisely, non-affine) $f: \mathbb{R} \to...
View ArticleDerivative of singular integrand
I am trying to differentiate this integral with respect to $x$:$$T(x,t) = {1\over\sqrt\pi} \int_0^t {g(s) \over \sqrt{t-s} } e^{-{x^2\over 4(t-s)}} ds$$According to this paper the derivative with...
View ArticleProve that family of holomorphic functions on unit disc is normal if...
Let $\mathcal{F}$ be a family of holomorphic functions on $\mathbb{D}=\{|z|<1\}$ so that for any $f \in \mathcal{F}$,$$\left|f^{\prime}(z)\right|\left(1-|z|^2\right)+|f(0)| \leq 1 \quad \text { for...
View ArticleContinuity of this function and well definedness
Consider $f: (0, 2] \to [-5, 7]$ defined as $f(x) = \begin{cases} x & 0 < x\leq 1 \\ 1 & 1 < x \leq 2 \end{cases}$Is this function well defined? Is it continuous?AttemptsI believe $f$ is...
View ArticleApplication of Stolz-Cesaro Theorem to a Series
I was given as an exercise to show that if $0 < s < 1$ and $\beta \geq 0$ then$$\lim_{n\to\infty} (1-s)n^\beta\sum_{k=1}^n\frac{s^{n-k}}{k^\beta} = 1.$$ Clearly $\beta = 0$ is a trivial case but...
View ArticlePuntual Convergence of a Series
Puntual Convergence of $\frac{\sin(x^{2n})}{1+nx^n}$?I think that The Ap Domain is $\Bbb R \setminus \{ \pi/2 + k\pi\}$ but I am not sure how I can prove it
View ArticleCan one show that the convolution operation and the bitwise multiplication is...
The foundation of this questions lies in an image processing question whether filtering and masking are interchangeable or not. I tried googling and calculating it from the integral- or...
View ArticlePartial Derivative of Pullback of Differential form
I'm new to differential forms and the book I'm reading contains a part I don't understand.It states the following:Let $k\geq 1$ and assume that $D \subset \mathbb{R}^k$ and $U \subset \mathbb{R}^n$ are...
View ArticleIs the product integral positive?
Let $f$ be a smooth function on $\mathbb{S}^1$ with $\int_{\mathbb{S}^1}f(x)\mathrm{d}x=0$. Can the integral$$\int_{0}^r\iint_{|x-y|<t}f(x)f(y)\mathrm{d}x\mathrm{d}y\mathrm{d}t$$be nonnegative for...
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Inequality involving exponentials and factorials
Let $b>1$ (a base), $n\ge 2$ and $1\le k\le n$. I would like to know for which $k$ the inequalities$$\frac{1}{n!}b^{n-1}\le \frac{b^k-1}{(n-k)!}\le\frac{1}{(n-1)!}(b^n-1)(b-1).$$hold or at least...
View ArticleTheorem 7.35 in Apostol's MATHEMATICAL ANALYSIS, 2nd edition: Is the...
Here is Theorem 7.35, in Chapter 7, in the book Mathematical Analysis - A Modern Approach to Advanced Calculus by Tom M. Apostol, 2nd edition:Assume $f \in R$ on $[a, b]$. Let $\alpha$ be a function...
View ArticleQuestion about proof of Lebesgue Decomposition Theorem for the case of...
I am asked to proof the following Lebesgue Decomposition Theorem for the case of $\sigma$-finite positive measure:Lebesgue Decomposition Theorem$\quad$ Let $(X,\mathscr{A})$ be a measurable space, let...
View ArticleTheorem 7.39 in Apostol's MATHEMATICAL ANALYSIS, 2nd ed: How to extract $g$...
Here is Theorem 7.39, in Chapter 7, in the book Mathematical Analysis - A Modern Approach to Advanced Calculus by Tom M. Apostol, 2nd edition:If $f$ is continuous on the rectangle $[a, b] \times [c,...
View ArticleLet $f\in L^p ([-1,1])$, then exists a continuous function $g$ on $[-1,1]$...
We know that if $\Omega$ is an open subset of $\mathbb{R}^N$, then the space of compact support functions $C_c(\Omega)$ is dense in $L^p(\Omega)$ for $p\in [1, \infty)$.Now, we suppose that $f\in...
View ArticleCounterexample: a function that is not log-Holder continuous
A function $p:\Omega\to\mathbb{R},\ \Omega\subset\mathbb{R}^N$ (open, bounded domain) is called log-Holder continous if:$$|p(x)-p(y)|\leq\dfrac{1}{-\ln(|x-y|)},\ \forall\ x,y\in\Omega,\ |x-y|\leq...
View ArticleA measurable bijection between the interval and the square
Is it possible to find a function $f:[0,1) \rightarrow [0,1)^2$ such that $f$ is bijective and $f$ as well as $f^{-1}$ are measurable with respect to the corresponding Lebesgue measures.If so how do I...
View ArticleFinding a weaker speed of divergance for a real sequance
Let\begin{equation*} \lim_{n} \frac{1}{n} \sum_{k=1}^{n-1} \frac{\ell_{k-1}}{\ell_{k}} \leq \gamma <1,~ and ~~ \frac{\ell_{k-1}}{\ell_{k}} \leq \frac{3}{2} ~ \forall ~ k \in...
View ArticleLogical consistency in proof for Real Cauchy sequence implies convergence...
I have doubts about this proof I reproduced from a text I have been following.Any help in full would be appreciated as it would really help me to get more familiar with the trickiness of analysis...
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