Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities. Do not use this tag just because an inequality appears somewhere in your question.

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Show that $|x_{(k)} - y_{(k)}| \leq \|x-y\|_2$

Given two vectors in $\mathbb R^n$, show that the absolute difference in the $k^{th}$ order statistics is bounded above by the $\ell_2$ norm. More formally, consider $x, y \in \mathbb R^n$. Show that: ...
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Lower bound on a particular infinite sum

Consider the following sum: $$\huge\sum_{k=1}^{\infty}{ A^{-k} B^{-\frac{\phi^k-1}{\phi-1}} }$$ where $\phi>1$, $A>0$ and $B>1$. The $B$ term dominates the $A$ term for large $k$, so the sum ...
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Asymptotics of a convoluted sum involving $r^{-\alpha}$

Let $\alpha,\beta$ be real constants satisfying $0<\alpha\leq \beta$, and let $R>1$ be an integer. Prove that \begin{equation}\label{eq:lemma_convolution} \sum^{R-1}_{r=1}\frac{1}{r^\beta (R-r)...
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Proving $(1-e^{4})(1-e^x) - (1-e)(1-e^{4x})> 0$ for all $0<x<1$.

I need help proving $(1-e^{4})(1-e^x) - (1-e)(1-e^{4x})> 0$ for all $0<x<1$. This question arrived from solving another question. I wanted to prove that a solution of an ODE was greater than $...
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the importance of inequalities tools in pure mathematics

I am a second year university student, my goal is to specialize in pure mathematics and for this reason I am currently trying to study calculus well through Spivak's book, and after finishing it I ...
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  • 111
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Solve $e^{-x} < \frac{1}{x^2 + 1}$ [duplicate]

I want to solve $e^{-x} < \frac{1}{x^2 + 1}$, assuming $x \in \mathbb{R}^+$. This can be solved via finding the solutions of $e^x>x^2+1$. This seems less troublesome, but I've found this ...
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$x_1x_2x_3x_4 > y_1y_2y_3y_4 \implies x_1+x_2+x_3+x_4 > y_1+y_2+y_3+y_4$

For strictly positive $x_i$ and $y_i$ I intuitively feel like the following should be true $$x_1x_2x_3x_4 > y_1y_2y_3y_4 \implies x_1+x_2+x_3+x_4 > y_1+y_2+y_3+y_4$$ but I can't seem to prove it ...
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Which one is the larger : $20!$ or $2^{60}$?

Which one is the larger : $20!$ or $2^{60}$ ? I am looking for an elegant way to solve this problem, other than my solution below. Also, solution other than using logarithm that uses the analogous ...
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Find the best constant $k$ such that the inequality is true .

It's a follow up of Prove that $\sum_{\mathrm{cyc}} \frac{214x^4}{133x^3 + 81y^3} \ge x + y + z$ for $x, y, z > 0$ : Problem : Let $x,y,z,k>0$ find the "best" value of $k$ such that : $...
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1 answer
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Inequalities concerning real positive numbers [duplicate]

Prove that $\frac{a}{b+c}+\frac{b}{c+d}+\frac{c}{d+a}+\frac{d}{a+b}\geq 2$ for all positive $a,b,c,d$ $a,b,c,d,$ are any positive real numbers
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Estimate $\int_{\mathbb R} f(z)\int_{z-1/2}^z dx\,\int_{\max(z,x+\epsilon)}^{x+1/2}\frac{1_{[a,b]}(x)1_{[c,d]}(y)}{(x-y)^2}dy$

How can I estimate the following expression $$\mathcal J:= \int_{\mathbb R} f(z)\int_{z-1/2}^z dx\,\int_{\max(z,x+\epsilon)}^{x+1/2}\frac{1_{[a,b]}(x)1_{[c,d]}(y)}{(x-y)^2}dy. $$ where $a,b,c, d \in \...
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Is it true that $\cos(\cos(1)) > \sin(\cos(1))$?

Let $\cos(1)$ be $\theta$. Then $\cos(\cos(1)) = \cos(\theta)$ and $\sin(\cos(1)) = \sin(\theta)$. We know that both $\cos(\theta)$ and $\sin(\theta)$ lies between $-1$ and $1$. What to do after this??...
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1 vote
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Show that $\frac{x \phi(x)}{2 \Phi(x) - 1}$ is decreasing for $x \geq 0$.

I am looking to prove that the following function $G: [0, \infty) \to \mathbb{R}$ is decreasing over it's domain: $$G(x) := \frac{x \phi(x)}{2 \Phi(x) - 1}$$ for $x \geq 0$. Here $\phi(x)$, and $\Phi(...
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2 answers
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Olympiad inequality $\frac{a}{c+5b}+\frac{b}{a+5c}+\frac{c}{b+5a}\geq \frac{1}{2}$ for all positive $a,b,c$?

How can I prove that $\frac{a}{c+5b}+\frac{b}{a+5c}+\frac{c}{b+5a}\geq \frac{1}{2}$ for all positive $a,b,c$?
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Proving inequality using root of unity

Let $\omega$ be a complex cube root of unity. It can be shown that if $a,b,c \in \mathbb{R}$, then $$(a+b+c)(a+b\omega+c\omega^2)(a+b \omega^2 + c \omega)= a^3+b^3+c^3-3abc$$ I was wondering if this ...
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Inequality using set

I am understanding the proof of a theorem below Theorem 1. Let $\mathbf{x}$ be s-sparse. Let $\eta>0$ be fixed. Suppose that $$ B_{\eta}(i) \cap B_{\eta}^{(2)}(j)=\emptyset, \quad \forall i, j \in \...
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Conjecture $\sum_{i=1}^{n}\sqrt{\frac{a_i^{2}}{a_{i+1}^{2}+a_{i}^{2}}}\leq n-1$ for $n\geq 4$ and $a_i>0$ where $a_{n+1}=a_1$

Problem/Conjecture : Let $a_i>0$ and $n\geq 4$ an integer then it seems we have : $$\sum_{i=1}^{n}\sqrt{\frac{a_i^{2}}{a_{i+1}^{2}+a_{i}^{2}}}\leq n-1$$ Where : $a_{n+1}=a_1$ I don't use numeric ...
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4 votes
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In proving $\int_{-x}^xf^\alpha\le f(-x)+f(x)$ for nonnegative continuous $f$, $\alpha\gt1$ implies $f\equiv0$, why can we take $f$ nondecreasing?

$\newcommand{\d}{\,\mathrm{d}}$I have a solution to this problem, but the TlDr is that my solution hinges on the assumption that $f$ is nondecreasing on $[0,\infty)$. It is obvious to me that proving ...
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1 vote
1 answer
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Making coefficients positive in a linear inequality constraint.

Suppose with have linear inequality constraints $q_i^Tx \leq c_i$ with $q_i \in \mathbb{Z}^n$ for $i=1,\ldots,m$ and $x \in \{0,1\}^n$. I would like to know if it is possible to reformulate $q_i^Tx \...
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0 answers
42 views

Continued fraction expression and inequalities

Given integers $0<a<n$ such that their gcd is 1, one can write $\frac{n}{a}=a_1-\frac{1}{a_2-\frac{1}{a_3-\cdots\frac{1}{a_k}}}$. Note that $a_i\geq 2$ and all $a_i \in \mathbb{Z}$. I have the ...
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  • 161
3 votes
2 answers
129 views

Prove that $\frac{a}{a^2+\lambda}+\frac{b}{b^2+\lambda}+\frac{c}{c^2+\lambda} \leq \frac{3}{\lambda +1}$

If $a+b+c=3$ ,$ a,b,c>0$ and $\lambda \geq 1$, prove that : $$\frac{a}{a^2+\lambda}+\frac{b}{b^2+\lambda}+\frac{c}{c^2+\lambda} \leq \frac{3}{\lambda +1}.$$ my attempt: using CBC twice and ...
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  • 111
1 vote
3 answers
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Why do you have to check all the 'sections' of an inequality when solving for x?

For instance, the inequality $x^2+3x+2>0$ factors into $(x+2)(x+1)>0$ If this were just an equation (i.e ...=0) you would know the solutions are x={-2, -1}. But, because it's an inequality, you ...
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2 votes
0 answers
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Furthest Distance Between Two Points on a Rectangular Prism

If the path between two points $x,y$ on a rectangular prism is the distance one would have to travel on the prism to reach point $y$ from $x$, what is the largest possible distance between two points ...
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2 votes
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Prove that $\prod\limits_{\mathrm{cyc}}\left(1+\frac{1}{\sqrt{ab}}\right)^a\geq 2^{a+b+c+d}$ for $a+b+c+d \le 4$

Let $a,b,c,d$ be positive real numbers satisfying $a+b+c+d\leq 4$. Show $$\left(1+\frac{1}{\sqrt{ab}}\right)^a\left(1+\frac{1}{\sqrt{bc}}\right)^b \left(1+\frac{1}{\sqrt{cd}}\right)^c\left(1+\frac{1}{\...
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1 vote
1 answer
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Proving $x^4+x^3y-x^2y+y^2>0$ with some given constraints

I have to prove the inequality $$x^4+x^3y-x^2y+y^2>0$$ holds for all pairs $(x,y)\in\mathbb{R}^2$ such that $x^2+y^2\leq1$ and $x^2>y^2$. I would like to find a lower bound just to start, but I ...
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1 vote
0 answers
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Another proof of Euler's inequality via the half-angle formulas

The Euler's inequality is an immediate consequence of Euler's identity in a triangle, $$OI^2=R^2−2Rr.$$ An additional proof of the Euler's inequality is given at Elias Lampakis, Am Math Monthly, 122 ...
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2 votes
2 answers
90 views

Integral involving the eror function

I would like to know if the following integral has an explicit expression or not. $$F(r)=\int_1^r e^{t^2/2} \operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)dt$$ where the error function inside is ...
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1 answer
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If $a,b,c,x,y,z$ are positive real numbers and $n>1$ then prove the following.

If $a,b,c,x,y,z$ are positive real numbers and $n>1$, prove that $(ax+by+cz)^n$ + $(ay+bz+cx)^n$ + $(az+bx+cy)^n$ 《 $(a+b+c)^n$($x^n$ + $y^n$ + $z^n$). I tired by Induction on n but got stuck on ...
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1 vote
1 answer
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Showing the maximum value using alternate approach

If $x_1,x_2...x_n$ are postive numbers satisfying $x_1\cdot x_2 \cdots x_n = 1$ , then find the maximum value of $\frac{1}{\sqrt{x_1 ^2 + 1}\,\cdot\, \sqrt{x_2^2 +1}\,\cdots\,\sqrt{x_n^2+1} }$ . ...
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0 votes
1 answer
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Proof inequality in $R^n$ [duplicate]

Let $x=(x_1,x_2,\dots,x_n).$ I wanted to show that $$\sum_{i=1}^{n}|x_i|-\sqrt{(\sum_{i=1}^{n}|x_i|^2)}\geq 0$$ I spent some times to show that but I am not sure if there are some inequalities that I ...
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4 votes
2 answers
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Find minimum of $\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2}+ab+bc+ca$

Let $a, b,c$ be the lengths of the sides of a triangle such that $a+b+c=2$, find the minimum value of $$\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2}+ab+bc+ca$$ I don't have many ideas for this problem,...
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1 vote
1 answer
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Some easy inequality for a new day :)))

I'll call expression of this form "innocent" $$\frac{M}{x^2+y^2+z^2} + \frac{N}{xy+yz+zx}$$ if we apply some inequality (like AM-GM, Bunyakovsky,...) and we still preserve the equality at $a=...
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3 votes
1 answer
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integral inequality of a non-increasing function on [0,1]... [duplicate]

I'm trying to prove an inequality from one of my real analysis books, and struggling. The problem is as follows: Show that for any non-increasing f: that maps [0,1] to the reals, and any θ in (0,1) we ...
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1 vote
0 answers
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Prove that given $V_{n}\sim\chi^2_{n}$. Then $\frac{V_{n}}{n}$ tends in probability to $1$ as $n\to\infty$. [closed]

Prove that given $V_{n}\sim\chi^2_{n}$. Then $\frac{V_{n}}{n}$ tends in probability to $1$ as $n\to\infty$. I´m doing this exercise and I does not know how to do it. Can anybody give me a hint? We ...
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4 votes
1 answer
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About a Conjecture: $-\left(\frac{x^{n}+1}{x^{n-1}+1}\right)^{n}-\left(\frac{x+1}{2}\right)^{n}+\left(x^{\frac{x}{x+1}}+\sqrt{x}-1\right)^{n}+1\leq 0$

Hi it's a conjecture wich refine for $0< x\leq 1$ the inequality Refinement of a famous inequality : Problem/Conjecture Let $0<x\leq 1$ then for $n\geq 3$ a natural number it seems we have the ...
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1 vote
1 answer
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Prove that $\sum_{n=0}^{\infty}\frac{|z|^n}{(2n)!}\leq\sum_{n=0}^{\infty}\frac{(|z|^\frac{1}{2})^n} {n!}$

I would like to ask how to prove the inequality: $\sum_{n=0}^{\infty}\frac{|z|^n}{(2n)!}\leq\sum_{n=0}^{\infty}\frac{(|z|^\frac{1}{2})^n}{n!}$ where $z\in\mathbb{C}$. I find this inequality in the ...
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0 answers
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Prove that $\sum_{k=1}^n\sqrt{x_k}\ge (n-1)\sum_{k=1}^n 1/\sqrt{x_k}$ [duplicate]

Given positive integers $x_1,...,x_n$ such that $$\sum_{k=1}^n\frac{1}{1+x_k}=1.$$Prove that $$\sum_{k=1}^n\sqrt{x_k}\ge (n-1)\sum_{k=1}^n \frac{1}{\sqrt{x_k}}.$$ This problem is from Problems from ...
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  • 3,141
5 votes
2 answers
76 views

Find the maximum of $\frac{abc}{(4a+1)(9a+b)(4b+c)(9c+1)}$,where$a,b,c>0$

$a,b,c>0$, find the maximum of : $$\frac{abc}{(4a+1)(9a+b)(4b+c)(9c+1)}$$ I try to find the minimum of $\frac{(4a+1)(9a+b)(4b+c)(9c+1)}{abc}=\frac{4a+1}{\sqrt{a}}\cdot\frac{9a+b}{\sqrt{ab}}\cdot\...
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  • 139
5 votes
5 answers
310 views

prove or disprove that:$a^2+b^2+c^2\leq \frac{27}{4}$

I tried to solve the below problem, I spend more than 5h just for prove it but without any result , this is the best attempt I did ,just I need to show that $a^2+b^2+c^2\leq \frac{27}{4}$ if $a,b,c&...
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  • 111
4 votes
2 answers
81 views

Is it possible to maximize $\frac{3t^2}{t^3+4}$ (where $t>0$) without taking derivative?

To find the maximum of $f(t)=\dfrac{3t^2}{t^3+4}$ (for $t>0$) we can simply equate the derivative with zero, $$f'(t)=0\Rightarrow 6t(t^3+4)-3t^2(3t^2)=0\Rightarrow -3t^4+24t=0\Rightarrow t=2$$ And $...
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  • 1,007
0 votes
0 answers
18 views

Converse Chebyshev inequality

The problem (and also the author´s solution): Of course I understand the derivation of the inequality etc. but I don´t see how this proves, that there is at least one entry of x with absolute value ...
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1 vote
0 answers
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Vector inequality regarding p-Laplace euations

In chapter~12, (p.99) of this note it is written that "In the case $p \geq 2$, the inequality $$|b|^p \geq |a|^p + p<|a|^{p−2}a, b − a> + C(p)|b − a|^p$$ holds with a constant $C(p) > 0$....
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3 votes
1 answer
35 views

Showing that $(|\left<a, b\right>| - \epsilon)^2 \leq |\left<a, Mb\right>|^2$ for self-adjoint operator $M$ such that $||(I - M)b|| < \epsilon$

Let $a, b$ be $L^2$ normalized functions and $M$ be a self-adjoint pseudodifferential operator such that $||(I - M)b|| < \epsilon$ and $\sigma(M) \leq 1$ (author of the paper I am reading has not ...
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1 vote
0 answers
44 views

Grover's Algorithm Trigonometric doubt

The Grover's operator $G$ in Quantum Computing, has the following effect $$ G|\Phi\rangle=(4\sin^2\Delta -1)|\Phi\rangle-2\sin\Delta|z\rangle $$ where $|\Phi\rangle=\frac{1}{\sqrt{2^n}}(|x_1\rangle+|...
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  • 6,853
-4 votes
0 answers
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Inequality - Simple. [closed]

How to show that for $a<0, x\in[0,1]$, we have: $e^{ax}-1-ax<\frac{1}{2} a^2x^2$? Please help me :/
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2 votes
2 answers
81 views

Prove that $x_1+2x_2+3x_3+\cdots+nx_n \leq \frac{n(n-1)}{2}+x_1+x_2^2+x_3^3+\cdots+x_n^n$ where $x_i > 0$ for all $i$ from $1$ to $n$ inclusive. [closed]

Question: Prove that $x_1+2x_2+3x_3+\cdots+nx_n \leq \frac{n(n-1)}{2}+x_1+x_2^2+x_3^3+\cdots+x_n^n$ where $x_i > 0$ for all $i$ from $1$ to $n$ inclusive. This is a nice problem from the polish ...
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2 votes
0 answers
58 views

Verification of an inequality involving $|x|^\alpha$

Let $u:\mathbb{R}^n\to \mathbb{R}$, $\alpha>0$ and $w(x)=|x|^\alpha.$ Then this paper I am reading has the following claim, $$\nabla^2\log w(\nabla u,\nabla u)+\frac{(\nabla \log w\cdot \nabla u)^2}...
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  • 8,806
2 votes
2 answers
58 views

Finding the set of parameters for which the inequality holds for all $x,y$

I encountered the following question about inequalities which I am curious how to solve. The simplest case is to consider the inequality $$|x|+|y|+|x+y|+ax+by\geq 0$$ where $x,y,a,b\in\mathbb{R}$. The ...
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-3 votes
1 answer
43 views

Is it true the following inequality? [closed]

Is it true the following statement: Let $R>0$, there exists $C_R>0$ such that $$\left |\int_0^T z(r) dr \right |^2\leq C_R \int_0^T \left | \int_0^\xi z(r)dr \right|^2 d \xi $$ for every $z \in ...
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0 votes
1 answer
53 views

Why does the following hold? $u_{k+1}\le(1+AT/N)u_k+BT^pN^{-p}$ implies $u_k\le e^{AT}u_0+\frac{B}{A}\frac{T^{p-1}}{N^{p-1}}(e^{AT}-1)$

I have read a lemma in "Numerical Integration of Stochastic Differential Equations" of G.N.Milstein, which was stated without any proof as follows: Suppose that for arbitrary natural number $...
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