Questions on proving, manipulating and applying inequalities.

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0answers
69 views

One-Sided Bivariate Chebyshev Inequality

Let $X$ and $Y$ be random variables with finite means $\mu_X$ and $\mu_Y,$ finite variances $\sigma_X^2$ and $\sigma_Y^2,$ and correlation $\rho.$ Let $A$ be the event that $X \leq \mu_X + k\sigma_X$ ...
0
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1answer
84 views

Error Term of Chebyshev inequality?

Chebyshev inequality tells us that $$Pr[|X-E[X]|\geq a]\leq \frac{Var[X]^2}{a^2}$$ Do you know an Expression (or a paper where this Expression is mentioned) for the error term?
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1answer
45 views

Prove that $\frac{\pi}{2}-x<\tan^{-1}(x)<\frac{\pi}{2}-x+\frac{x^3}{3}$

Prove that for every $x>0$, it is true: $$\frac{\pi}{2}-x<\tan^{-1}(x)<\frac{\pi}{2}-x+\frac{x^3}{3}$$ We can split it into two statements: $\frac{\pi}{2}-x<\tan^{-1}(x)$ ...
23
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1answer
622 views

Is $\pi$ the best constant in this inequality?

Let $E$ be the set of completely monotonous functions on $[0,+\infty)$, that is $f \in C^\infty([0,+\infty))$ and $\forall\, n\geq 0,\forall\, x\geq 0,\quad(-1)^nf^{(n)}(x)\geq 0.$. For $f\in E$ and ...
2
votes
2answers
44 views

Calculate the inequality

$a,b,c$ are the sides of a triangle.Then show that $$(a+b+c)^3 > 27(a+b-c)(b+c-a)(c+a-b)$$ Also give the case where equality holds i.e. $$(a+b+c)^3=27(a+b-c)(b+c-a)(c+a-b)$$ I tried triangle ...
1
vote
1answer
36 views

Inequality concerning sides of a triangle

I am trying to prove the following inequality: take $x,y\in \mathbb{R}^N\setminus\{0\}$ with $|x|\le |y|$ then $$\frac{1}{4}|x-y|\le |t x+(1-t)y|,\ \forall\ t\in [0,1/4].$$ I could prove some ...
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1answer
30 views

Application of the Hoelder inequality

How to prove using Hoelder Inequality that, $$\sum\limits_{i=1}^n \mathbb{E} (O (|X_i| + |X_i|^3 )) \leq n \, \mathbb{E} (O ( |X|^3 ) ),$$ where $X = (X_1, X_2, \ldots X_n)$ are i.i.d. independent ...
0
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2answers
188 views

Intuition or figure for Reverse Triangle Inequality $||\mathbf{a}| − |\mathbf{b}|| ≤ |\mathbf{a} − \mathbf{b}|$ (Abbott p 11 q1.2.5)

I acquiesce to Wikipedia's picture for Triangle Inequality. But without referring to Triangle Inequality at all, is there intuition or figure please for Reverse Triangle Inequality for all ...
2
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1answer
75 views

Elementary matrix inequality

Let $A \in \mathbb{R}^{n \times n}$ be a positive semidefinite matrix. Is the mapping $$ \begin{align} F \ \colon \ \mathbb{R}^{n \times n} &\to \mathbb{R}^{n \times n} \\ X &\mapsto X^{-1} - ...
2
votes
2answers
37 views

Prove $(|OP|)+ |PQ|)^2 > |OQ|^2$

I did all the algebra and for some reason I'm getting 0 > $y_2^2$ which is clearly wrong. Where did I mess up at?
5
votes
3answers
136 views

How prove this inequality $\sum_{k=1}^{n}\frac{2k-1}{k\binom{n}{k}}\ge \frac{n}{2^{n-1}}$

let $1\le k\le n,k,n\in N^{+}$, show that $$\sum_{k=1}^{n}\dfrac{2k-1}{k\binom{n}{k}}\ge \dfrac{n}{2^{n-1}}$$ I know this $$\sum_{k=1}^{n}(2k-1)=n^2$$ and $$\sum_{k=1}^{n}k\binom{n}{k}=n\cdot ...
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1answer
102 views

Changes in the hypotheses of a mean-value theorem

For $X \subset \mathbb{R}^d$ open, we define $$ C^1(X) := \left \{ f : X \to \mathbb{C} : f \text{ is a function s.t. } \frac{\partial f}{\partial x_j} \text{ exists and is continuous for } j = 1, 2, ...
2
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0answers
124 views

Proof of integral inequality

How does one prove (without making use of any approximations whatsoever) the following inequality: $$\int_1^2 \left(\ln(x)\right)^{2013}dx\leq\dfrac{1}{2^{2013}}.$$
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0answers
38 views

Simplyfying inequality.

Given that $a = x + y$ $b = y + z$ $c = z + x$ Is there a way to simplify $a^2b(a-b)+b^2c(b-c)+c^2a(c-a) \ge 0$ to $x^3y + y^3z + z^3x \ge x^2yz + xy^2z + xyz^2$ without just simply ...
1
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1answer
48 views

Plotting a complex inequality

So I am looking at the exercise: "Plot $|z-3i| + |z-4| > 7 $ in the complex plane." I have done similar exercises by using $z = x + iy $ and treating the problem as a real valued inequality, but ...
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2answers
53 views

Logarithmic equation

I'm studying logarithms and I encountered this equation: $$[\log_9(k+1)]^2+\log_9(k+1)+(k+1)>3$$ I tried a lot but I still couldn't solve it! I know this may be easy for most of you but please ...
5
votes
2answers
145 views

How to prove $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$ [duplicate]

Let $f$ be $C^1$ in $[-\pi, \pi]$ and satisfies $\int_{-\pi}^\pi f(x)dx=0$, periodic boundary condition. Then, prove that $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$. I try to prove ...
4
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1answer
195 views

How prove this $100+5(a^2+b^2+c^2)-2(a^2b^2+b^2c^2+a^2c^2)-a^2b^2c^2\ge 0$

let $a,b,c\ge 0$, and such $$a+b+c=6$$ show that $$100+5(a^2+b^2+c^2)-2(a^2b^2+b^2c^2+a^2c^2)-a^2b^2c^2\ge 0$$ My idea: since $$a^2+b^2+c^2=(a+b+c)^2-2(ab+bc+ac)=36-2(ab+bc+ac)$$ ...
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1answer
37 views

If F is real-entire, then how to write, $F(z)- F(w)$ in terms of $(z-w)$ and $(\bar{z}- \bar{w})$?

Define $F:\mathbb C \to \mathbb C$ such that $F(z)= \sum_{j,k=0}^{\infty}c_{j,k} z^{j} \bar{z}^{k}$ is an entire real analytic function on $\mathbb C$ with $F(0)=0.$ My question is :How to show: ...
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2answers
83 views

number of positive integer solution of inequation

Given an inequation with P,Q,R all integers, $P \cdot R \cdot b + P \cdot Q \cdot c - Q \cdot R \cdot a \geq 0$ how many positive integer solutions of $(a, b, c)$ ? Here $a \leq P, b \leq Q, c \leq ...
0
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1answer
40 views

Establish a relation between p and q

For positive real numbers $a_1, a_2, ... ,a_{100}$, let $$p = \sum_{i=1}^{100} a_i $$ and $$q = \sum_{1\le i \lt j \le 100} a_ia_j \space .$$ Then establish an inequality(or equality) among $p$ and ...
0
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2answers
49 views

How to solve this equation - $\sqrt{3-x}\geq 1+\sqrt{x}$

$$\sqrt{3-x}\geq 1+\sqrt{x}$$ The second square $(\sqrt{x})$ give me problems. Solution must be: $$0\leq x <\dfrac{3-\sqrt{5}}{2}$$
2
votes
2answers
38 views

Parametric inequation…

Supppose we have $a$ a real positive number that's not equal to $1$. Solve the following inequation: $$\log_a(x^2-3x)>\log_a(4x-x^2)$$ If it's known that $x=3.75$ is one solution of it.
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5answers
2k views

Very tough inequality I cannot prove

Suppose $x \geq 0 $ is real number, then $$ x^x \geq \sqrt{ 2x^x - 1 } $$ How can I show this? Also, what is the greometrical implication of such inequality ? thanks
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1answer
37 views

What is the domain of $Z=\sin(\ln(x\,\arccos{y}))$?

What is the domain of $Z=\sin(\ln(x\,\arccos{y}))$? I see that is should be $-\dfrac{\pi}{2}\leq \ln(x\,\arccos{y}) \leq \dfrac{\pi}{2}$ and then $e^{-\dfrac{\pi}{2}} \leq x*\arccos(y) \leq ...
0
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3answers
78 views

How to prove that $\log\left(\frac{x+1}{x}\right) - \frac{1}{x+1}$ is always positive and tends to zero?

What I get so far is the inequality: $$1 + \frac{1}{x} > e^{\frac{1}{x+1}}$$ which if we expand: $$1+\frac{1}{x} > 1+\frac{1}{x+1}+\frac{1}{2! (x+1)^2}+\cdots$$ and I cannot prove this ...
2
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4answers
94 views

Show $\frac{b}{c} + \frac{c}{a} + \frac{a}{b} \ge 3$ for $a,b,c > 0$.

Disclaimer: The statement may be false, but for now I'm operating under the assumption it's true and trying to prove it. My workings: I got a common denominator and expressed it as: $$\frac{ab^2 + ...
1
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0answers
155 views

Relation between determinant and L1 norm

Recently, I have coped with a problem about the relation between determinant of positive definite matrices and their L1 norm. More specifically, assume that $\Sigma_{1}$ and $\Sigma_{2}$ are two ...
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3answers
47 views

What is the domain of $z=\arcsin\dfrac{x}{y}$?

I get that it should be $|y|>|x|$ and in the Wolfram it looks like this. But when I graph it by hand is that it should be only the "upper" part of intersection and not the "bottom" part as well, ...
3
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2answers
64 views

An inequality for some series

Consider real positive numbers $t_1,t_2,\cdots, t_n$ for some $n\in\Bbb N$, with $\sum_{i=1}^nt_i^2=n$, such that if $0<t_i<1$ then $$\frac{t_i}{\sin\left(\frac{t_i\pi}{1+t_i}\right)}<1$$ ...
3
votes
3answers
482 views

Maximum value of $abc$ for $a, b, c > 0$ and $ab + bc + ca = 12$

$a,b,c$ are three positive real numbers such that $ab+bc+ca=12$. Then find the maximum value of $abc$
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0answers
217 views

Can proof by contradiction and counterexample by used at the same proof?

Here is a part of a theorem: If $\alpha>1$ and $x\ge-1$ then $(1+x)^\alpha \ge 1 + \alpha x$ I was wondering if I could use proof by contradiction and counterexample at the same time. Assume ...
1
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1answer
67 views

One inequality involving Total variation function

If $F$ is of bounded variation in $[a,b]$, then I need to prove that $$ \int_{a}^{b}|F'(x)| dx \leq T_F(a,b)$$ If $F'$ were Riemann integrable then it was easy to prove (in fact we can prove ...
4
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1answer
73 views

How prove this there exist positive integer sets $A$ such $|A+A|>|A-A|$

let positive integer $n\ge 8$ is given, show that: there exist sets $A$ with the set of is positive integer.such $|A|=n$,and such $$|A+A|>|A-A|$$ where $$A-A=\{a-b|a\in A,b\in A\},A+A=\{a+b|a\in ...
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3answers
226 views

Not sure how to find the limit of this inequality?

I'm trying to solve the limit of this inequality. The question goes as follows: If $$4x - 9 \leq f(x) \leq x^2 - 4x + 7$$ for $x \geq 0$, find $\lim_{x\to 4} f(x)$. I'm not really sure how to go ...
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1answer
30 views

Finding radius of convergence

I have gotten the problem almost solved, but I'm hung up on how to solve this inequality: $$|x|/|2x+1|<1 $$ I could move the denominator to the right side of the equation: $$|x|<|2x+1|$$ But ...
0
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1answer
65 views

Prove that if $n\geq1$ and $a_1, a_2, \ldots, a_n$ are any real numbers, then $|a_1 + a_2 +\ldots+ a_n| \leq |a_1|+ |a_2| +\ldots+ |a_n|$

I understand that if all values of $a$ are positive, then $|a_1 + a_2 +\ldots+ a_n| = |a_1|+ |a_2| +\ldots+ |a_n|$. I also understand that if any values of a are negative, then $|a_1 + a_2 +\ldots+ ...
3
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1answer
120 views

Prove that inequality holds for all real number from $[0,1]$

There are given real numbers $a_1,a_2, ... , a_n \in [0,1]$ Prove that $\displaystyle \sum_{1\le i\le n} a_i \le 1+ \sum_{1\le i} \sum_{<j\le n} a_ia_j$ I have problem here because I can't find ...
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4answers
64 views

$(n+1)!>n^2, \forall n\ge 4$

The base case is clear, since if $n\gt 4, 5!=120\gt 25=5^2$ So assume $n=k$ which shows that $k!\gt k^2$. Then if $n=k+1$, $$(k+1)!=(k+1)k!$$ $$\gt (k+1)k^2 $$ Induction argument $$=k^3+k^2$$ $$\gt ...
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1answer
546 views

proof following Bessel's inequality and Fourier series

Letting $a_n$ denote the coefficients in the Fourier cosine series of any function $f(x)$ on $(0,\pi)$ , how can I show that: $\sum_{n=1}^N c_n^2 \le ||f∥^2$, where $c_n$ are the Fourier constants ...
2
votes
3answers
168 views

Range of modulus of Complex Number

If $z\in \mathbb{C}$ and $$ |z-1|+|z+3|\le 8$$ Find the Range of $$|z-4|$$ My Try: $$|2z-8|=|2z+2-10|\le |2z+2|+10=|(z-1)+(z+3)|+10\le|z-1|+|z+3|+10\le18$$ $\implies$ $$|z-4|\le9$$ I need Hint to ...
3
votes
1answer
102 views

How prove this inequality $\frac{1}{n}\sum_{k=0}^{n-1}\binom{k}{a}\binom{k}{b}\le\frac{1}{a+b+1}\binom{n}{a}\binom{n}{b}$

let $a,b,n$ be positive integer numbers,and such $a,b\le n$, show that $$\dfrac{1}{n}\sum_{k=0}^{n-1}\binom{k}{a}\binom{k}{b}\le\dfrac{1}{a+b+1}\binom{n}{a}\binom{n}{b}$$ this inequality maybe ...
2
votes
0answers
108 views

Proving AM-GM via $n \cdot (a_1^n + a_2^n + \dots + a_n^n) \ge (a_1^{n-1} + a_2^{n-1} + \dots + a_n^{n-1}) \cdot (a_1 + a_2 + \dots + a_n)$

I want to prove the arithmetic–geometric mean inequality. To prove that, I need the following inequality: Suppose that $n$ is an integer which is greater than or equal to $1$ and $a_1, a_2, \dots, ...
1
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3answers
80 views

How to solve this algebraically

This seems like a really stupid question to ask here... I'm trying to solve $\sqrt{x^2 - 1} + x > 0$. When I try this happens: $\sqrt{x^2 - 1} > - x$ $x^2 - 1 > x^2$ (squared both sides) ...
0
votes
1answer
107 views

About the Gronwall inequality

If I have that $$||\eta_u(t)||\leq 1+C_1\int_0^t \frac{1}{||\eta(s)||}||\eta_u(s)||ds$$ and $$\sqrt{1-\frac{2\varepsilon}{C}}||u||\leq ||\eta(s)||\leq 2||u||$$ how to obtain using the Gronwall ...
3
votes
3answers
119 views

Proving $\cos 36° > \tan 36° $

How do we prove that $\cos 36° > \tan 36° $ ? Please help . Thank you.
0
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2answers
40 views

Inequation of an sum smaller than 1

I'm trying to figure out the following $$ \sum^{\infty}_{n=3} \dfrac{q!^2}{n!^2} < 1 $$ How I can show it if $q \geq 2$? Maybe with telescoping sums? Thanks, Landau
3
votes
1answer
65 views

Determining equality of sets defined by polynomial inequalities in several variables

Let's say I have two sets $S_1,S_2 \subseteq \mathbb{R}^n$ each defined by a number of polynomial inequalities. Is there a computationally feasible way to find whether $S_1 = S_2$? In particular, is ...
1
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2answers
39 views

Prove the inequality $(a^2+1)/(b+c)+(b^2+1)/(a+c)+(c^2+1)/(a+b)\ge 3$

If $a,b,c\in\mathbb R^+$ prove that $(a^2+1)/(b+c)+(b^2+1)/(a+c)+(c^2+1)/(a+b)\ge 3$.
7
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1answer
308 views

How to prove this sequence of inequalities

The number $c_{g}(n)$ is defined by the recurrence \begin{equation} c_{g}(n) = c_{g}(n-1)+ (n-1)(n-2)c_{g-1}(n-2) , \end{equation} with $c_{0}(n)=1$ for any $n\geq 1$ and $c_{g}(n)=0$ if $n \leq 2g$. ...