Questions on proving and manipulating inequalities.

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25 views

generalized mean inequality related problem

Let $x_1\leq x_2\leq \ldots\leq x_k\leq y\leq x_{k+1}\leq\ldots \leq x_n$. Let $p>1$. Does the following inequality is true? ...
2
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0answers
163 views

An improvement of Jensen's inequality - help please!

It would nice if someone could help me with this problem. I am looking at an improvement to the classical Jensen's Inequality: $$\int_\limits{}^{} \phi(x) \mu \mathrm{d}x \geq ...
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3answers
45 views

Why doesn't inequality hold as a property in natural number induction?

It is said that all natural numbers follow the rule of induction: if a said property holds for one number and for its successor, it holds for all natural numbers. But, let us define the following ...
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22 views

Inequalities of quadratic form

We know that the below inequality holds if A is positive definite ${\lambda _{\min }}\left( {{A}} \right){\left\| x \right\|^2} \le {x^T}{A}x$ or equivalently $\alpha{\left\| x \right\|^2} \le ...
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1answer
37 views

Easier way to prove $|x_1y_1-x_2y_2|\leq|x_1-x_2|+|y_1-y_2|$

I have to prove that for $(x_1,y_1),(x_2,y_2)\in[0,1]^2$ $$|x_1y_1-x_2y_2|\leq|x_1-x_2|+|y_1-y_2|$$ Now what I do is this: w.l.o.g. say $y_1<y_2$. Def ...
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2answers
30 views

Complex Analysis Modulus Inequality

I am working on a proof, and I'm stuck with one inequality. I am given that for all $z$ (and large $z$ in particular) and $g$ an entire function, we have $$|g(z)|\leq\sqrt{|z|}+1/\sqrt{|z|}$$ Now ...
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3answers
32 views

Disproving a inequality implication by contradiction.

Let $x,y \in R$. If $0 \leq y < x$ for all $x > 0$, then $y=0$. Proof by contradiction: Assume the opposite that is; "If $0 \leq y < x$ for all $x > 0$, then $y\neq0$". ...
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31 views

(Riesz's lemma) A closed subspace of a Banach space

Let V be a Banach space over R Let W be a proper closed subspace of V Prove : For any $\epsilon > 0$, there is a v $\in$_V_ such that ||v||=1 and ||v+w||$\geq$ $1$ - $\epsilon$ And my proof ...
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1answer
29 views

Banach space and its closed subspace. a vectors satisfying inequality.

V=a Banach space over R W=a proper closed subspace of V Prove : For any $\epsilon > 0$, there is a v $\in$_V_ such that ||v||=1 and ||v+w||$\geq$ $1$ - $\epsilon$ I have shown that there exists ...
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2answers
41 views

Inequality for a random variable

Let $\xi > 0$ be a random variable such that $\mathbb E[\exp{(-\xi)}]=\exp{(-a)}$. How to show that for any $c>0$ $$P(\xi \ge c) \le \frac{1-\exp{(-a)}}{1-\exp{(-c)}}$$
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1answer
25 views

vector norms involving quantities with a filter relationship

I have two vector norm quantities: $\|\Psi^T(t)\Theta(t)\|$ and $\|\Phi^T(t)\Theta(t)\|$. Here $\Phi^T(t),\Psi^T(t)\in \mathbb{R}^{m\times n}$ and $\Theta(t)\in\mathbb{R}^{n}$. There is a filter ...
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4answers
41 views

Is the difference of two decreasing functions also decreasing?

If I have two decreasing functions $f(x)$ and $g(x)$, is their difference $ H(x)=f(x)-g(x) $ also a decreasing function?
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1answer
72 views

Proof by induction: For all $n \geq 1$; $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots +(-1)^{n+1} \frac{1}{n} \leq 1$

Proof by induction: For all $n \geq 1$; $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots +(-1)^{n+1} \frac{1}{n} \leq 1$ This is what I have so far: Base case: for $n = 1$ ...
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0answers
34 views

Squaring an inequality?

I want to sample a mouse moving around the screen (sending mouse samples over the network) proportional to acceleration, so that samples are more dense when curvature is higher, as well as when change ...
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3answers
36 views

Doubt in application of AM-GM inequality

The Question is What is the minimum value of $9\sec^2 x + \cos^2 x$ ? Now, I tried applying AM-GM inequality, and the answer comes out as 6. But sec2x has a minimum value of 1. So the least value ...
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1answer
71 views

Find minimum of $P=\frac{\sqrt{3(2x^2+2x+1)}}{3}+\frac{1}{\sqrt{2x^2+(3-\sqrt{3})x +3}}+\frac{1}{\sqrt{2x^2+(3+\sqrt{3})x +3}}$

For $x\in\mathbb{R}$ find minimum of $P$. $P=\dfrac{\sqrt{3(2x^2+2x+1)}}{3}+\dfrac{1}{\sqrt{2x^2+(3-\sqrt{3})x +3}}+\dfrac{1}{\sqrt{2x^2+(3+\sqrt{3})x +3}}$ Source : Viet Nam national test for high ...
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0answers
16 views

Minimizing sum of cubes givien some constraints

I am given a sequence of real numbers $b_1 \ge b_2 \ge \cdots \ge b_n$ and two positive integers $m > k.$ Let $x_2 \ge \cdots \ge x_{m}$ be numbers such that $$b_{n-m+i} \le x_i \le b_{i} \quad ...
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1answer
19 views

Vector inequality for a scalar difference of two vectors in $\mathbb{R}^n$.

A student posed an interesting problem to me the other day and embarrassingly I could not prove or disprove it even though it appears relatively simple. The question was: Given vectors ...
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0answers
31 views

Application of Jensen's Inequality. Correct?

Help would be appreciated. Consider $x \in (0,1)$ and $f(x)=x^2$ which is convex we want to show that $\mathbb{E}\Big[f(X)\Big] \geq f\Big[\mathbb{E}(X)\Big]$. Therefore, $\mathbb{E}\Big[X^{2}\Big] ...
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0answers
95 views

Let $p$=prime and $\sqrt{x}+\sqrt{y}<\sqrt{2p}$

Let $p$ be a fixed odd prime. Let $x,y\in \mathbb{Z}_+$ such that $\sqrt{x}+\sqrt{y}<\sqrt{2p}$. Prove that $$\sqrt{x}+\sqrt{y}\le \sqrt{\frac{p-1}{2}}+\sqrt{\frac{p+1}{2}}.$$ Any ideas at all? ...
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0answers
16 views

A gronwall inequality

In Majda/Bertozzi book, Incompressible flows etc.. p.118,he uses Gronwall theorem on the following inequality: $$|\nabla v(.,t)|_{L^{\infty}} \leq C\left( 1 + \int_0^t|v(.,s|_{L^{\infty}}ds ...
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1answer
56 views

Estimating $\left|\frac{1}{y} - \frac{1}{y_0}\right|$ when $|y-y_0|$ is small [closed]

If $|y-y_0| < \min\left(\frac{|y_0|}{2}, \frac{\varepsilon |y_0|^2}{2}\right)$ where $y\neq 0$ and $y_0 \neq 0$, Prove that $\left|\frac{1}{y} - \frac{1}{y_0}\right| < \varepsilon$. I was ...
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2answers
49 views

What can we say about $x$?

If we have three reals (not positive reals, just reals) $x,y,z$ such that $x^2+3y^2+z^2+(x+y-z)^2=2$, what can we say about $x,y,z$? Is it possible to find the minimum of $x$? I don't know where to ...
5
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2answers
108 views

Which term of the binomial expansion of $\left(1+\sqrt{2}\right)^{50}$ is the greatest?

Which term of the binomial expansion of $\left(1+\sqrt{2}\right)^{50}$ is the greatest? How can I find it, without comparing all 51 values? Is there a quicker way to do it? (The solution says ...
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0answers
47 views

Inequality with complex number

Let $z,z'\in\mathbb{C}$. I want to prove that $$\vert\vert z\vert^{\alpha}z-\vert z'\vert^{\alpha}z'\vert\leq C (\vert z\vert^{\alpha}+\vert z'\vert^{\alpha})\vert z-z'\vert$$ where $\alpha$ is an ...
11
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1answer
78 views

Valid proof of Young's Inequality?

Part of an exercise to prove Holder's inequality in Rudin involves proving Young's Inequality... That is, given $\frac{1}{p}+\frac{1}{q} = 1$, prove $$ab \leq \frac{a^p}{p} + \frac{b^q}{q}.$$ ...
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1answer
34 views

Proving statements about ceiling and floor functions.

Prove or disprove the statements below. (a) For all positive real numbers x and y, $\lfloor x \cdot y\rfloor ≤ \lfloor x\rfloor \cdot \lfloor y\rfloor $. (b) For all positive real numbers x and y, ...
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3answers
63 views

Proof by induction: $\log n < n$ for $n ≥ 1$.

I was just wondering it is possible to prove this statement via mathematical induction? (I know you can do it via calculus but I want to specifically do it via induction). I have given it a go but am ...
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1answer
84 views

Impossible to prove that is unbounded!

How we can demonstrate that $$\int _1^e\:\left(1+\log\left(x\right)\right)^ndx$$ is unbounded as $n\to \infty$, without using Bernoulli's Inequality?
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1answer
37 views

How do I integrate the inequality $ \frac{f(\frac{1}{2}+h)+f(\frac{1}{2}-h)}{2} \leqslant f(\frac{1}{2})$ over the range $h\in[0,1/2]$?

I would like to know the formal steps and theory. I was told that, by integrating this inequality, I can achieve one of the definitions of a concave function in the interval [0,1]. Thanks for your ...
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17 views

Prove a lower bound $\left|\int_{-\infty}^\infty k(t) g(t) e^{i\theta t}dt\right|\geq C \int_{-\infty}^\infty |f(t)|dt$

Let $k(t)$ be any function absolutely integrable over $(-\infty,\infty)$ and let $$g(t)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(u) e^{itu}du$$ Consider $$\int_{-\infty}^\infty k(t) g(t) e^{i\theta t}dt$$ can ...
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2answers
67 views

Prove that $\sqrt{2011} + \sqrt{2013} + \sqrt{2015} + \sqrt{2017} < 4 \sqrt{2014}$

How to prove that $\sqrt{2011} + \sqrt{2013} + \sqrt{2015} + \sqrt{2017} < 4\sqrt{2014}$ without using calculator?
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1answer
44 views

Direct comparison test for ( Improper ) Integrals [closed]

How we can prove with direct comparison test for ( Improper ) Integrals that is bounded: $\int _1^n\:e^{-x^3}dx$ ?
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2answers
76 views

Inequality very difficult to show

1) $\int _0^1\:\frac{x^n}{x^n+1}dx\ge \int _0^1\:\frac{x^{n+1}}{x^{n+1}+1}dx$ but I dont want to use $I_{n+1}-I_n$ 2) How we can prove with direct comparison test for ( Improper ) Integrals that is ...
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3answers
52 views

Does $a \leq b + c $ imply $a^2 \leq (b+c)^2 + (b-c)^2$?

Givens $$ a \leq b + c $$ or $$ a^2 \leq b^2 + c^2+2bc $$ Can we prove that?: $$ a^2 \leq (b+c)^2 + (b-c)^2 = 2b^2 + 2c^2 $$
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1answer
38 views

Very difficult to prove a convergent with Weierstrass

How we can prove that is monotone and bounded: $I_n=\int _1^n\:e^{-x^3}dx\:$ , Have any ideea how we can solve? and explain all to understand, I am a student... P.S: for all guys on this site, you ...
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1answer
94 views

Fourier series for convex plane curves.

The following problem is from Stein's Fourier analysis. This problem explores another relationship between the geometry of a curve and Fourier series. The diameter of a closed curve $\Gamma$ ...
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4answers
64 views

$\sum_{k=1}^n \log k \ge \int_1^n \log x \, dx$

Why is $$\sum_{k=1}^n \log k \ge \int_1^n \log x \, dx$$ is there an intuitive or graphical way to think about it?
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1answer
33 views

How to write Holder's inequality for random vectors?

For $1 < p,q < \infty$ satisfying the constraint $1/p + 1/q =1$ and for $X, Y$ random variables such that $\mathbb E [\vert X \vert ^p ], \mathbb{E} [\vert X \vert ^q ] < \infty $ we have ...
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2answers
72 views

How to prove that the $\lceil x y \rceil \le \lceil x\rceil\lceil y\rceil$ for real numbers $x, y$?

I know intuitively that this is true, but whenever I try and break apart that intuition to see where it's coming from, I essentially end up re-writing the assumption I'm trying to prove. I've tried ...
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1answer
62 views

Estimating partial sums $\sum_{n = 1}^m \frac{1}{\sqrt{n}}$

Apostol's Calculus, exercise number I 4.7 13. Prove that if $n \geq 1$, then $$ 2(\sqrt{n+1} - \sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n} - \sqrt{n-1}) $$ and use this to prove that if ...
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0answers
29 views

On the existence of a certain sequence of positive numbers II

I wish to find a sequence of strictly positive real numbers $(a_1, a_2, \dots)$, such that $$ \sum_{k = 1}^\infty \frac{a_k}{k} < \infty $$ and such that for all $m, n \in \{1, 2, \dots\}$ with $m ...
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0answers
35 views

About Riemann's hypothesis and a comment by 'almagest'.

When I was user 128932 I asked if a and b are relatively prime and are both exceptions to Robin's inequality then so is ab. The user 'almagest' confirmed this I think. So if N is sufficiently large ...
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0answers
30 views

Proving Inequality $\sum_{m=1}^{M}\beta_m\ln(x_m)\geq 0 $

We have $\beta_m=\dfrac{x_m^{M-2}}{\prod_{j=1,j \neq m}^{M} (x_m-x_j)}$ Hence, it is easy to show that $\sum_{m=1}^{M}\beta_m = 0$ However, I am unable to show that $\sum_{m=1}^{M}\beta_m\ln(x_m) ...
5
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5answers
251 views

How to write an expression in an equivalent form without absolute values?

The question I have in front of me is the very first problem in Trench's Introduction to Real Analysis: Write the following expression in equivalent form not involving absolute values: $a+b+|a-b|$ ...
1
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2answers
29 views

Inequality with absolution value for complex number

How to show that inequality: $|1-\bar{\alpha} z| \ge |z-\alpha|$ $z$ and $\alpha$ are complex number, $\alpha$ is constans and $|z|<1$, $| \alpha| < 1$ I can proof that by using substition ...
0
votes
1answer
24 views

Why does the unit vector of form $x_i=\frac{-1}{\sqrt{n}}$ minimize sum of $x_i$?

Cauchy-Schwarz implies that for $||\vec{x}||=1, \vec{y}=(1,\ldots,1)\in\mathbb R^n,\sum_{i=1}^{n} x_i = \pm\sqrt{n}$ if $\vec{x}=\pm{k}\vec{y}$. This implies that ...
0
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0answers
27 views

Simple number theory inequality

$$-b < -r \leq 0\text{ and } 0 \leq r' < b \implies -b < r'-r < b$$ how is that implication possible? I'm going over the proof for the division theorem mainly the uniqueness part ...
2
votes
1answer
22 views

Show $\sum_{n\leq k\leq 2n}2^{-2k}\log(k)\leq C\, 2^{-2n}\log(n)$

I'd like to prove $$\sum_{n\leq k\leq 2n}2^{-2k}\log(k)\leq C \, 2^{-2n}\log(n),$$ where $C>0$ is a constant. Can someone give me a hint.
2
votes
1answer
29 views

Combinatorial sum inequality

Prove the following inequality: $$ \forall k\in\left\{4n+5:n\in\mathbb{N}\right\},\qquad\sum_{m=0}^{\frac{k-1}{2}}{\left( -1 \right) }^{m}\binom{k}{2m}2^{2m}\neq 1. $$ I'm particularly interested ...