Questions on proving and manipulating inequalities.

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1
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0answers
19 views

search on a split data structure

I have the following problem: Part 1: Lets say I have n items in a data structure and I want to search for them. I know that a subset of my data $r \cdot ...
0
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0answers
22 views

An inequality of probability [closed]

Suppose $X$ is a random variable. $E(X^2)=1$, $E|X|\ge a>0$. Let $\lambda\in[0,1]$, prove that $$P(|x|\ge\lambda a)\ge(1-\lambda)^2a^2.$$ I have tried to use Chebyshev's inequality, but I didn't ...
-1
votes
0answers
25 views

Quadratic recurrence inequality

I have the recurrence relation: $r_{k+1} \leq r_k^2+ (1/2)r_k \quad (k =1,2,\ldots)$, where each $r_k$ is non-negative and $r_1<1$. I have the following questions in this regard: A simple plot ...
-1
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1answer
28 views

an inequality for multiplication of cubic numbers

I need to know is there a positive constant $c$ such that $$ |a_1 \cdots a_n|^3 \leq c\big(|a_1|^3+\cdots+|a_n|^3\big), $$ where $a_{i}\neq 0$ ? I tried geometric and arithmetic inequalities but I ...
1
vote
2answers
44 views

Concluding three statements regarding $3$ real numbers.

$\{a,b,c\}\in \mathbb{R},\ a<b<c,\ a+b+c=6 ,\ ab+bc+ac=9$ Conclusion $I.)\ 1<b<3$ Conclusion $II.)\ 2<a<3$ Conclusion $III.)\ 0<c<1$ Options By ...
6
votes
2answers
141 views

Find the minimum value of $A=\frac{2-a^3}{a}+\frac{2-b^3}{b}+\frac{2-c^3}{c}$

Let $a, b$ and $c$ three positive real numbers such that $a+b+c=3$. Find the minimum value of $$A=\frac{2-a^3}{a}+\frac{2-b^3}{b}+\frac{2-c^3}{c}.$$ Here is my attempt. By symmetry we can assume that ...
5
votes
1answer
75 views

Inequality with reciprocals of $n$-variable sums

Let $a_1,a_2,\ldots,a_n$ be positive real numbers. Is it always true that $$\sum_{i=1}^n\frac{1}{a_i}-\sum_{1\leq i<j\leq n}\frac{1}{a_i+a_j}+\sum_{1\leq i<j<k\leq ...
2
votes
3answers
41 views

Reasoning about numbers close to two other numbers $a,b$ (inequalities)

Let $a < b$ and $0 < \varepsilon < (b - a)$ and let $x, y \in \mathbb R$ be such that $$ | x - a | < \frac{(b - a) - \varepsilon}{2}, \qquad | y - b | < \frac{(b - a) - ...
0
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1answer
33 views

An inequality involving logarithmic function and floor function [closed]

How do I show that $${\log_2(\log_2(x))}\geq\lfloor{\ln(\ln(x))}\rfloor.$$
0
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3answers
56 views

Proving that $|a-b|≤|a|+|b|$ [closed]

Can someone prove this to me: $$|a-b|≤|a|+|b|$$ I am in 8th grade and I have this for my homework. Thanks for helping.
3
votes
2answers
112 views

Show that $\frac{(x^2 + y^2 )}{4} \leq e^{x+y-2}$

Show that \begin{equation} \frac{x^2 + y^2}{4} \leq e^{x+y-2} \end{equation} is true for $x,y \geq 0$. As far, I have prove that \begin{equation} x^2 + y^2 \leq e^{x}e^{y}\leq e^{x+y} ...
19
votes
4answers
190 views

Inclusion-exclusion-like fractional sum is positive?

Let $A_1,A_2,\ldots,A_n$ be finite nonempty sets. Is it true that $$\sum_{i=1}^n\frac{1}{|A_i|}-\sum_{1\leq i<j\leq n}\frac{1}{|A_i\cup A_j|}+\sum_{1\leq i<j<k\leq n}\frac{1}{|A_i\cup ...
2
votes
1answer
131 views

If $a,b,c>0, a+b+c=3$, minimize $\frac{2-a^3}{a}+\frac{2-b^3}{b}+\frac{3-c^3}{c}$ [duplicate]

Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Find the minimum value of the expression $A= \frac{2-a^3}{a}+\frac{2-b^3}{b}+\frac{3-c^3}{c}$ I tried solving it, but I got nothing
2
votes
1answer
41 views

Minkowski's inequality

Minkowski's inequality for sums states $$\left(\sum_{j=0}^\infty |a_j+b_j|^2 \right)^{1/2} \le \left(\sum_{j=0}^\infty |a_j|^2 \right)^{1/2}+\left(\sum_{j=0}^\infty |b_j|^2 \right)^{1/2} $$ for ...
1
vote
2answers
63 views

Minimum value of cosA+cosB+cosC in a triangle ABC

I have used jensen's inequality but couldn't move on.
1
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5answers
83 views

Proving that $\frac{1}{a^2}+\frac{1}{b^2} \geq \frac{8}{(a+b)^2}$ for $a,b>0$

I found something that I'm not quite sure about when trying to prove this inequality. I've proven that $$\dfrac{1}{a}+\dfrac{1}{b}\geq \dfrac{4}{a+b}$$ already. My idea now is to replace $a$ with ...
4
votes
2answers
115 views

How to prove this maximum is $\frac{\sqrt{3}}{5}$

let $a,b,c>0$, such $ab+bc+ac=1$,show that $$\dfrac{1}{3a+5b+7c}+\dfrac{1}{3b+5c+7a}+\dfrac{1}{3c+5a+7b}\le\dfrac{\sqrt{3}}{5}$$ since dear Mac sir,he solve with inequality ...
2
votes
3answers
35 views

$\frac{1+\sqrt x}{1-\sqrt x} \leq e^{Cx}\frac{1+x}{1-x}$

I try find a constant $C$ such that $$\frac{1+\sqrt x}{1-\sqrt x} \leq e^{Cx}\frac{1+x}{1-x}$$ Maybe is not posible
1
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0answers
22 views

The decay of the Fourier coefficients of the disjoint union of arcs

Given $N$ disjoint arcs $\{I_{\alpha}\}_{\alpha=1}^{N}\subset\mathbb{T} $,set $f=\displaystyle\sum_{\alpha=1}^{N}\chi_{I_{\alpha}}$ show that $$\sum_{|v|>k}|\hat{f}(v)|^2\le\dfrac{CN}{k}$$ This ...
0
votes
4answers
50 views

How to isolate $x$ in the inequality $3^{x+2}<5^{x-1}$

For example, \begin{align} 0.4^x &\gt 0.9 \\ \left(\frac{2}{5}\right)^x &\gt \left(\frac{9}{10}\right) \\ \left(\frac{2}{5}\right)^x &\gt \left(\frac{2}{5}\right)^{\log_{2/5} ...
5
votes
3answers
141 views

Prove that $A_{100} \gt 14$ where $A_{n}=A_{n-1}+\frac{1}{A_{n-1}}$ and $A_1=1$

I tried attempting the question, and the best upper bound I could obtain was $1+\ln{98}$. I tried using $A_{n}\le n$ to form a harmonic series, but that wasn't strong enough. Any help would be ...
3
votes
1answer
56 views

Complex Inequalities

Let $\mathbb{H}=\{z\in \mathbb{C} | \ \Im(z)>0\}$ and $f:\mathbb{H} \to \mathbb{H}$ analytic. Prove that for every $z_1, z_2 \in \mathbb{H}$, it must happen that $$ ...
2
votes
1answer
39 views

Proving $1-\cos(k)\geq\frac{2}{\pi^{2}}k^{2}$ for $k\in(-\pi,\pi)$

I am trying to prove the following: $$1-\cos(k)\geq\frac{2}{\pi^{2}}k^{2}\quad\hbox{for}\quad k\in(-\pi,\pi]$$ So far, I have tried using some Maclaurin expansion arguments, but when that didn't ...
0
votes
1answer
49 views

3 variable symmetric inequality

Show that for positive reals $a,b,c$, $\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\geq \frac{3a^3+3b^3+3c^3}{2a^2+2b^2+2c^2}$ What I did was WLOG $a+b+c=1$ (since the inequality is homogenous) ...
1
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4answers
29 views

How to express a=8 versus b=4?

I want to know how to express in English a and b. a is larger(=greater, bigger) than b as four. (right?) b is smaller(=less) than a as four. (right?) thank you for reading my question.
2
votes
5answers
450 views

Find the maximum possible area of a certain right triangle

I want to find the maximum possible area of a right triangle with hypotenuse $=10$. My approach so far: let $x,y$ be the lengths of the two sides adjacent to the right angle; then $$100=x^2+y^2$$ ...
0
votes
1answer
19 views

How are the following inequalities concluded based on this first one?

$$I-\frac{\epsilon}{3} \leq s(f,T) \leq \underline{I} \leq \overline{I}\leq S(f,T) \leq I+ \frac{\epsilon}{3}$$ from this, the following is concluded, but how? $$1.\ \ \ 0 \leq |I-\underline{I}|\leq ...
4
votes
2answers
40 views

Given $a_1 \ge \cdots \ge a_n$ and $b_1 \ge \cdots \ge b_n$, then show $\sum a_ib_{\pi(i)}$ is maximum when $\pi=id$.

Suppose $a_1 \ge \cdots \ge a_n$ and $b_1 \ge \cdots \ge b_n$ are two sequences of positive real numbers. Then show $\sum a_ib_{\pi(i)}$ is maximum when $\pi=id$. Here, $\pi \in S_n$. I ...
1
vote
1answer
17 views

Matrix factorization inequality

How does one show that the following matrix factorization inequality holds in $M_{n} (\mathcal{A})^{+}$, $$(a_{i}^{*}a^{*}aa_{j}) \leq ||a^{*}a|| \cdot (a_{i}^{*}a_{j})$$ Notation. Let $M_{n} ...
2
votes
2answers
59 views

$xy + yz + zx + 2xyz = 1$ implies $4x+y+z\geq 2$

Let $x,y,z>0$ satisfy $$xy + yz + zx + 2xyz = 1.$$ Prove that $4x+y+z\geq 2$. The condition invites the factoring $(1+x)(1+y)(1+z)+xyz-2=x+y+z$, but having the factor $4$ in the desired inequality ...
-1
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1answer
47 views

Does $xy > xz$ imply $y > z$? [closed]

$xy > xz$ We don't know the sign of $x$. Can we conclude $y>z$ from above?
4
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0answers
65 views

About two combinatorial counting problems.

Here are the problems: Suppose $X$ is a set of $n$ elements, and $S_1,...,S_m$ are $m$ subsets of $X$ of average size at least $n/w$. Show that if $m\geq 2kw^k$, then there are $k$ distinct ...
1
vote
1answer
52 views

Estimation of a probability of marginal values of a random variable

My question is related with this question on combinatorics of 0-1-matrices from MO. Trying to obtain a (asymptotic) lower bound for $\alpha(n)$ by probabilistic approach (see, for instance, “The ...
5
votes
2answers
361 views

Prove the inequality using AM-GM inequality

Given that $a,b,u,v \geq 0$ and $$a^5+b^5 \leq 1$$ $$u^5+v^5 \leq 1$$ Prove that $$a^2u^3+b^2v^3 \leq 1$$ This looks like Holder's inequality, but I found this problem in a book just after the AM-GM ...
3
votes
2answers
55 views

$1+\sqrt[3]{e^{2a}}\sqrt[5]{e^{b}}\sqrt[15]{e^{2c}} \leq \sqrt[3]{(1+e^{a})^2}\sqrt[5]{1+e^{b}}\sqrt[15]{(1+e^{c})^2}$

The inequality $1+\sqrt[3]{e^{2a}}\sqrt[5]{e^{b}}\sqrt[15]{e^{2c}} \leq \sqrt[3]{(1+e^{a})^2}\sqrt[5]{1+e^{b}}\sqrt[15]{(1+e^{c})^2}$ is true for all $a,b,c\in\mathbb{R}$? I've tried to use the ...
2
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2answers
45 views

Verification for this proof

Sorry guys about the verification questions but it's near the end of the semester and I am very sheepish about making mistakes especially because real analysis is a very important course it's only the ...
2
votes
1answer
51 views

Sum of Square-Weights

For positive reals $a,b,c$, prove that $$\frac{a^3+b^3+c^3}{3abc}+\sum_{\text{cyc}} \frac{a(b+c)}{b^2+c^2}\geq 4$$ I've heard of a lemma stating if a polynomial expression $f(a,b,c)$ satisfies both ...
1
vote
2answers
38 views

Nesbitt's Inequality for 4 Variables

I'm reading Pham Kim Hung's 'Secrets in Inequalities - Volume 1', and I have to say from the first few examples, that it is not a very good book. Definitely not beginner friendly. Anyway, it is ...
2
votes
6answers
103 views

If $f'$ is increasing and $f(0)=0$, then $f(x)/x$ is increasing

Let $a>0$ and $f:[0,a] \to \mathbb{R}$ continuous function that is twice differentiable on $(0,a).$ Also $f(0)=0$ and $f'$ is strictly increasing function on $(0,a).$ I have to show that the ...
2
votes
1answer
19 views

Strict inequality linked with sequence

Suppose a sequence $a_{n}$ is defined in following way: $a_{1}=1, a_{n}=a_{1}a_{2}a_{3}…a_{n-1}+1, n \geq 2$. Prove, that for every natural number $m$ an inequality holds ...
4
votes
3answers
49 views

Prove the inequality $4+xy+yz+zx \ge 7xyz$

Let $x,y,z$ be non negative real numbers such that $x+y+z=3$, prove the following inequality: $$4+xy+yz+zx \ge 7xyz$$ I tried MV and taking out one variable but I got nothing
1
vote
2answers
94 views

Valid AM-GM inequality proof?

We have to prove that $$\frac{(x_1+x_2+x_3+...+x_n)}{n} \geq (x_1\cdot x_2\cdot x_3\cdots x_n)^{1/n}$$ Attempt: Raising both sides to the nth power gives ...
6
votes
0answers
74 views

Could this approximation be made simpler ? Solve $n!=a^n 10^k$

I need to find the smallest value of $n$ such that $$\frac{a^n}{n!}\leq 10^{-k}$$ in which $a$ and $k$ are given (these can be large numbers). I set the problem as : solve for $n$ the equation ...
1
vote
1answer
33 views

Solve the inequality $x^2+y^2 \geq\frac{1}{x+y}+2(x-y -1).$

For which real values $x,y$ does the inequality $x^2+y^2 \geq\frac{1}{x+y}+2(x-y -1)$ hold? I have found for $x=\dfrac{1}{10}, y=\dfrac{3}{10}$ that LHS=RHS but cant solve it in the general ...
4
votes
4answers
148 views

How to prove that $\sum_{1}^{\infty} \frac{1}{n^3} \le 1.5$

I have this sequence: $$\sum_{1}^{\infty} \frac{1}{n^3}$$ and I need to prove that: $\sum_{1}^{\infty} \frac{1}{n^3} \le 1.5$ So basically I know that this sequence converges using the integral ...
0
votes
2answers
15 views

The Cauchy-Schwarz inequality for Hermitian forms

Let $V$ be a vector space over a field $\mathbb K$ and $f$ be a nonnegative Hermitian form on $V$. Then, $\forall x,y \in V$: $$|f(x,y)|^2 \le f(x,x)f(y,y)$$ Here's one proof: For an ...
0
votes
1answer
22 views

For given mean $\mu$ of random variable X in [0,1], what is the probability distribution function $p(X)$ that makes $VAR(X)$ maximum?

Given the conditions $\int_{0}^{1} p(x)dx=1$, $\int_{0}^{1} xp(x)dx=\mu$ and $p(x)\ge0$ for $\forall x \in [0,1]$, What probability distribution function $p(x)$ makes $Var(X)$=$\int_{0}^{1} ...
1
vote
1answer
39 views

A question on Hölder inequality [duplicate]

Let $p, q > 1$ such that $\frac{1}{p} + \frac{1}{q} = 1$. Then $$|\sum\limits_{i = 1}^n x_i y_i| \leq ||x||_p ||x||_q, \;\; \forall x, y \in \mathbb{R}^n.$$ I have to prove it considering $$u = ...
2
votes
3answers
44 views

Maximum value of trigonometric expression [closed]

If $r=3+\tan c \tan a, \quad q=5+\tan b \tan c, \quad p=7+\tan a \tan b$ Provided $a,b,c$ are positive and $a+b+c=\dfrac{\pi}2$ Find the maximum value of $\sqrt p + \sqrt q + \sqrt r$ .
0
votes
1answer
16 views

About Chebyshev inequality for integrals

Let $u \in H^1(\Omega) \cap C(\Omega)$ ($\Omega \subset R^n$ a smooth and bounded domain) a nonnegative function. Let $B(x,R) \subset \overline{ B(x,R) } \subset \Omega $ a ball. Suppose that ...