Questions on proving and manipulating inequalities.

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Trouble simplfying quadratic indentity

I'm struggling to follow the derivation below: $u_j=\frac{\sigma^4+ \theta^2\delta_j^2\alpha_j^2+2\theta\sigma^2\delta_j\alpha_j}{\sigma^2+\delta_j^2\alpha_j^2} \leq \frac{\sigma^4+ ...
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1answer
15 views

Normal distribution tail probability inequality

I am trying to show that $P(X>t)\leq (1/2)\exp(-t^2/2)$ for $t>0$ where $X$ is a standard normal random variable. Perhaps this is simple. I have been starting with $$ \int_{t}^{\infty} ...
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1answer
25 views

Prove $\sqrt{ \frac{2x^2 - 2x + 1}{2} } \geq \frac{1}{x + \frac{1}{x}}$ for $0 < x < 1$

I stumbled upon this question while doing practice inequalities questions, and I do not know how to start... Problem: Prove that \begin{align*} \sqrt{ \frac{2x^2 - 2x + 1}{2} } \geq \frac{1}{x + ...
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3answers
106 views

Solve $\frac{x^2+2xy+y^2}{x^2-y^2} >x+y$

Find the set of integer solutions $(x,y)$ to $$\frac{x^2+2xy+y^2}{x^2-y^2} >x+y$$ I can't seem to multiply both sides by the expression in the denominator. Nor can I simplify and cancel any ...
2
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1answer
29 views

Prove the Inequality on Prime Counting Function

Is there any way to prove that, $$\pi(x^2)-\pi(y^2) \geq \sqrt{\pi(x-y)}$$ I have tried to prove it using inequalities on $\pi(x)$ but it didn't work. Can anyone help me?
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3answers
46 views

Inequality: Find Min $S=\frac{a}{\sqrt{1-a}}+\frac{b}{\sqrt{1-b}}$

Inequality: Find Min a,b>0, a+b=1. $S=\frac{a}{\sqrt{1-a}}+\frac{b}{\sqrt{1-b}}$
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1answer
20 views

Inner product inequality

Is there any inequality for $x^TAy$ where $x$ and $y$ are vectors and $A$ is positive definite matrix. For example: $x^TAy\ge k||x||||y||$ where $k$ is a coefficient of (min or max) eigenvalue of ...
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2answers
34 views

Show numbers are positive

Given the following set of equations, show that all the variables (for any length list) must each be $\ge 0$. $$ 2x_1 -x_2 \ge 0 $$ $$ -x_1 + 2x_2 -x_3 \ge 0 $$ $$ -x_2 + 2x_3 -x_4 \ge 0 $$ $$ -x_3 + ...
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0answers
33 views

How can I find the complex numbers satisfying this condition?

For a given complex number $a$ with $|a|\ge1,$ I want to find the all complex numbers on the unit circle such that $$\dfrac{z}{(a-z\bar a)^2}\in\mathbb{R}$$ and satisfying the condition ...
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1answer
23 views

Two inequalities imply third?

I'm spinning my wheels here - I was able to figure out the solution to my previous question: Constructing Lyapunov function for system of ODEs, but need help on the finishing touch. $$\mathrm{Does} \ ...
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0answers
21 views

A question about the definition of Lipschitz continuity

Suppose $f(x,y)$ is a function on $R^2$. If $f$ is Lipschitz continuous with respect to $y$, then $|f(x,y_1)-f(x,y_2)|<C|y_1-y_2|$ for some constant $C$. But can anyone tell me whether the ...
1
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1answer
35 views

How to show this inequality in the law of iterated logarithm

Let $\psi(t) = \sqrt{2t\log\log t}$ and $q>4$. Then we have for large $k$, $$ \psi\left(q^k - q^{k-1}\right) \geq \psi\left(q^k\right) \left(1-\frac{1}{q}\right). $$ To prove this, I can tell by ...
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1answer
23 views

between what two disjoint sections we can do a unification in order to get this group of solutions?

between what two disjoint sections we can do a unification in order to get this group of solutions? $0<|x+6|\leq{0.4}$ in other words, in what values should I fill the blankets: (____,____) ...
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0answers
71 views

An elementary annoyance

I'm going through some notes and I'm having problem understanding an inequality: The objects involved are: $X$ is a real-valued random variable with mean zero. We consider $n$ identical copies ...
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0answers
22 views

If $f^\alpha$ is Lipschitz, what can be said about $f$?

If $f$ is a positive function and $f^{\alpha}$ with $0<\alpha\leq1$ is Lipschitz then what we can say about $f$? Ps: I need to know what is the exponent holder continuity of $f$.
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1answer
29 views

Holder Inequality

I have a problem with the demonstration of this inequality ('m following Royden) : If $p$ and $q$ are nonnegative numbers such that $\frac{1}{p}+\frac{1}{q}$ and if $f \in L^p$ and $g \in L^q$, then ...
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0answers
13 views

Bounding the inverse of a diagonally dominant matrix entry-wise

I have a $d \times d$ matrix $A$ whose entries are bounded (C1): $I - \epsilon X \preceq A \preceq I + \epsilon X$, where $I$ is the identity matrix and $X = 11^\top - I$ is the matrix with ones ...
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1answer
20 views

Is this inequality true

Let $a_k, \lambda_k>0$, $k=1, \ldots, n$ such that $\sum_{k=1}^n\lambda_k=1$. Is it true $\left(\sum_{k=1}^n\lambda_ka_k\right)^2\left(\sum_{k=1}^n\lambda_k/a_k^2\right)\ge 1$?
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1answer
27 views

Proving (a part of) Hoeffding's lemma

Hoeffding lemma goes like this: *Let $X$ be a scalar variable taking values in an interval $[a,b]$. Then for any $t>0$ $$\mathbb{E} e^{tX}\leq e^{t\mathbb{E} ...
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0answers
17 views

Logarithm inequality theoretical problem

I just want to ask why is it that the inequality sign reverses when we take antilog of an in equation. I understand it a bit practically but is there a proof? Like if log(f(x))< log(g(x)) Then ...
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17 views

Question regarding previous elementary number theory question.

Regarding the question here: James has 773500 gold coins to purchase a number of hats and ties..... I solved the question all the way up until I got the complete solutions of: x = 416500 - 16n y = ...
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0answers
28 views

Condition for trigonometric inequality

I want to prove the following statement: Suppose $\frac{1}{4}(\cos(\theta_1)+\cos(\theta_2))^2+\lambda^2(a\sin(\theta_1)+b\sin(\theta_2))^2\leq 1$ holds for all $\theta_1,\theta_2\in[-\pi,\pi]$, then ...
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5answers
42 views

Proof of $\arcsin x \le 2\arctan x$?

I am looking for a proof for the following 'fact': $$ \arcsin x \le 2\arctan x \quad \forall x\in[0,1). $$ I put fact between single quotes, as the only proof I found is a plot by wolframalpha. I know ...
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0answers
9 views

Maple command for finding specific values that will make an inequality true

Is there a Maple command for finding specific values that will make an inequality true? For instance, if the input is $a<b,$ a possible output would be $a=1,b=2.$
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1answer
69 views

Proof of $p_n<n^2$ by Elementary Means

Is there any proof of the inequality $p_n<n^2$ (for all sufficiently large $n$) by elementary means and without using Prime Number Theorem? I searched in google but in vain. The results that I ...
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2answers
26 views

Is $\frac{\vert x+y\vert}{1+\vert x+y\vert}\leq\frac{\vert x\vert}{1+\vert x\vert}+\frac{\vert y\vert}{1+\vert y\vert}$ true?

Is $\frac{\vert x+y\vert}{1+\vert x+y\vert}\leq\frac{\vert x\vert}{1+\vert x\vert}+\frac{\vert y\vert}{1+\vert y\vert}$ true for all $x,y\in\mathbb{R}$? If not, how can I prove that $\int\frac{\vert ...
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3answers
52 views

How is using inequalities $\ge$ and $\le$ different for being used to solve equations?

I seriously want to know the difference for finding the solution to an equation and an inequality using $\ge$ and $\le$. I know how to solve inequalities involving variables like $9x+5$ $\ge$ $42$, ...
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4answers
36 views

prove that this equality is always right for each positive x and y.

prove that this inequality is hold for each positive x,y. $x\over\sqrt{y}$ + $y\over\sqrt{x}$ $\ge$ $\sqrt{x}$ + $\sqrt{y}$ I want a detailed way of solving the question.
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0answers
16 views

Find inequality for gaussian density

Let $C>0$ be a fixed constant. Is it true that $$Cx^2 e^{-x^2}\leq e^{-\frac{x^2}{C}}?$$ More generally, if we have a power $x^p$ in front of the exponential, do we have that $$(C^{1/2}x)^p\leq ...
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0answers
23 views

Problem in proving norm

I have a linear space V that includes the continuous functions from [-$\pi, \pi$] to the complex set C, of the form: $$f(x) = a\cos(t)+b\sin(t) $$ where $a$ and $b$ are complex numbers. I want to ...
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3answers
88 views

Prove the inequality$\int_0^{+\infty} {\sin x \over x}dx<\int_0^\pi {\sin x \over x}dx$

Prove: $$\int_0^{+\infty} {\sin x \over x}dx<\int_0^\pi {\sin x \over x}dx$$ Here is my answer,but I want a different way to prove it. \begin{aligned} \int_0^{+\infty} {\sin x \over ...
4
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2answers
62 views

An inequality with $a_n=\int_0^1 \frac{\mathrm{d}x}{\underbrace{\sqrt{2+\sqrt{2+\dots+\sqrt{2x}}}}_{n \text { times}}}$

Let the sequence $(a_n)_n$ defined by $$a_n=\int_0^1 \frac{\mathrm{d}x}{\underbrace{\sqrt{2+\sqrt{2+\dots+\sqrt{2x}}}}_{n \text { times}}}$$ 1)Prove that $$\frac12 \leq a_n \leq ...
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5answers
53 views

Prove that $e^x \ge$ its Maclaurin polynomial with n terms [on hold]

a) show that $e^x \geq 1+x$ for all $x\geq 0$ b) deduce that $e^x \geq 1+x+\frac{1}{2}x^2$ for $x\geq0$ c) use induction to prove that for $x\geq 0, n\in \mathbb{N}$ $$e^x\ge ...
1
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1answer
10 views

Find all of the exact solutions of the equation and then list those solutions which are in the interval [0, 2pi)

This is for trigonometric equations and inequalities: Find all of the exact solutions of the equation and then list those solutions which are in the interval [0, 2pi) Cos(9x)=9
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2answers
52 views

Help me to prove this inequality. [on hold]

$$\text{Let } a,b\in R , a\neq 0. \text{ Show that } a^2+b^2+\frac{1}{a^2}+\frac{b}{a}\ge \sqrt{3}.$$
3
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1answer
30 views

Prelude to Cauchy-Schwarz, Quadratic proof.

I have a problem in trying to prove the following observation: "Show that if $ a,b,c \in \mathbb{R} $ are such that for all $ \lambda \in \mathbb{R} $, $a\lambda^2 + b\lambda +c \geq 0 $ then $ b^2 - ...
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2answers
19 views

Prove that the image of $f: (0, \infty) \to R$ is contained in $[2, \infty)$. [on hold]

where $f(x) = x + 1/x$ Any help is appreciated, what I did was completely wrong haha..
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2answers
36 views

How to estimate $\sum_{n=0}^\infty (\log\log2)^n/n! $ from below?

How to show that the following inequality holds: $$\sum_{n=0}^\infty \frac{(\log\log2)^n}{n!}>\frac 35$$ Is it possible to prove this using induction?
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votes
4answers
50 views

How to solve a convoluted absolute value inequality?

$$ \lvert \lvert x-2\rvert -3\rvert \lt 5 $$ How can I attack this the best way? I see that both sides are positive. Squaring yields: $$ \lvert x-2\rvert ^2 -6 \lvert x-2\rvert +9\lt 25 $$ $$ ...
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1answer
27 views

An exponential inequality

Assume $a(t)\geq 0$ and $b(t)\geq 0$. i can show the following inequality $\mid e^{-\int_0^ta(s)ds}-e^{-\int_0^tb(s)ds} \mid\leq T\max_{0\leq t \leq T}\mid a(t)-b(t)\mid$ by writing $\mid ...
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2answers
78 views

minimize a function using AM-GM inequality

I want to minimize the function $$ \frac{x}{1-x^2} + \frac{y}{1-y^2} + \frac{z}{1-z^2} $$ subject to the constraint $$x^2 + y^2 + z^2 = 1 \space\text{and} \space x,y,z > 0$$ Wolfram Alpha tells ...
5
votes
3answers
128 views

Proof of Nesbitt's Inequality?

I just thought of this proof but I can't seem to get it to work. Let $a,b,c>0$, prove that $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge \frac{3}{2}$$ Proof: Since the inequality is homogeneous, ...
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2answers
67 views

Proving $|P(A\cap B)-P(A)P(B)|\leq \frac{1}4$

Let $A$ and $B$ be two events of a probability space. Prove that $\displaystyle|P(A\cap B)-P(A)P(B)|\leq \frac{1}4$ I think it's a very challenging problem, and I've made no progress so far ... ...
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1answer
39 views

How to prove this inequality relating to trigonometric function?

In a triangle, A, B, C are three corners of the triangle, try to prove that : $$\root 3 \of {1 - \sin A\sin B} + \root 3 \of {1 - \sin B\sin C} + \root 3 \of {1 - \sin C\sin A} \geqslant {3 \over ...
4
votes
5answers
121 views

Showing that $e^{-2} < \ln 2$

I have to prove the following inequality: $e^{-2} < \ln2.$ Using Bernoulli's inequality, I showed that $2 \leq e$, and using this result I tried to simplify the inequality by using an upper ...
5
votes
3answers
256 views

Beautiful cyclic inequality

Prove that cyclic sum of $\displaystyle \sum_{\text{cyclic}} \dfrac{a^3}{a^2+ab+b^2} \geq \dfrac{a+b+c}{3}$ , if $a, b, c > 0$ I'm really stuck on this one. Tried some stuff involving QM> ...
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3answers
47 views

Prove that: $\sqrt[3]{a+2b}+\sqrt[3]{b+2c}+\sqrt[3]{c+2a}\le 3\sqrt[3]{3}$

Given $a,b,c>0$ and $a+b+c=3$. Prove that: $\sqrt[3]{a+2b}+\sqrt[3]{b+2c}+\sqrt[3]{c+2a}\le 3\sqrt[3]{3}$
-3
votes
3answers
71 views

Prove that: $\sqrt{4-a^2}+\sqrt{4-b^2}+\sqrt{4-c^2}\le 3\sqrt{3}$ [closed]

if $0<a,b,c\le2$ . Prove that: $\sqrt{4-a^2}+\sqrt{4-b^2}+\sqrt{4-c^2}\le 3\sqrt{3}$
0
votes
0answers
21 views

Machine Floating Point Theorem

Completely stuck on this floating point question. Let $x \in \mathbb{R}$ have the following floating point representation: $$ x = (-1)^s[0.a_1a_2\dots a_ta_{t+1}\dots]\cdot \beta^e $$ [Where $\beta$ ...
0
votes
2answers
25 views

Inequalities with expected value on one side and probability on the other

In a part of a proof I am following, the author states that $$\displaystyle \mathbb{E}\left[\frac{|X_n - X|}{1 + |X_n - X|}\right] \leq \epsilon + \mathbb{P}(|X_n - X| > \epsilon)$$ and ...