Questions on proving and manipulating inequalities.

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0
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28 views

How find this minimum of the $|PA_{1}|+|PA_{2}|+|PA_{3}|+\cdots+|PA_{n}|$

Question: give the $n$ point $$A_{1}(x_{1},y_{1}),A_{2}(x_{2},y_{2}),A_{3}(x_{3},y_{3}),A_{4}(x_{4},y_{4}),\cdots,A_{n}(x_{n},y_{n}),x_{i}\in R,y_{i}>0$$ Find a ponit $P(x,0)$,such ...
1
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3answers
40 views

Does x + y have a maximum value under the following conditions?

$ x ≥ 0$, $ y ≥ 0$, $2x + y < 8$ $x + 2y < 10$ Does x + y have a maximum value under the above conditions? How I tried to do it: I knew that x and y are positive numbers, and if trying to ...
0
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0answers
26 views

For any real number $p \geq -1$ and any positive $n$, $(1+p)^n\geq1+np$ [duplicate]

How can I prove this: For any real number $p \geq -1$ and any positive $n$, $(1+p)^n \geq 1+np$. I don't have any idea how to start.
11
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4answers
297 views

Suppose $xyz=8$, try to prove that $\sqrt{\frac{1}{1+x}}+\sqrt{\frac{1}{1+y}}+\sqrt{\frac{1}{1+z}}<2$

Who can help with the following inequality? I can prove it but using some rather ugly approach (e.g. by leveraging the derivative of $\frac{1}{\sqrt{t+1}}+\frac{1}{2}\sqrt{1-\frac{8}{t^2+8}}$ to show ...
0
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0answers
53 views

Equality case in Hölder's inequality

How can I show that $$\left(\int{p(x)^{1-\sigma}\mathrm dx}\right)^{\frac{1}{1-\sigma}}\cdot \left(\int y(x)^\frac{\sigma-1}{\sigma}\mathrm dx\right)^{\frac{\sigma}{\sigma-1}}=\int p(x) ...
1
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1answer
121 views
+50

How prove this inequality $\sqrt{(2a+1)^2+(2b-\frac{\sqrt{3}}{3})^2}+\sqrt{(2a-1)^2+(2b-\frac{\sqrt{3}}{3})^2}+\cdots$

Question: let $a,b\in R$, show that $$\sqrt{(2a+1)^2+(2b-\dfrac{\sqrt{3}}{3})^2}+\sqrt{(2a-1)^2+(2b-\dfrac{\sqrt{3}}{3})^2}+\sqrt{4a^2+(2b+\dfrac{2\sqrt{3}}{3})^2}\ge ...
2
votes
4answers
52 views

Show that $ax^2+2hxy+by^2$ is positive definite when $h^2<ab$

The question asks to "show that the condition for $P(x,y)=ax^2+2hxy+by^2$ ($a$,$b$ and $h$ not all zero) to be positive definite is that $h^2<ab$, and that $P(x,y)$ has the same sign as $a$." Now ...
1
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1answer
82 views

Inequality: $\left|x^3-y^3\right|<|x|^3+|y|^3$

Could anyone show me why $$\left|x^3-y^3\right|<|x|^3+|y|^3$$ for all real numbers (x,y) except 0? I'm thinking of whether of how to remove the modulus sign on the left hand side of the ...
4
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6answers
435 views

The process of solving the inequality $\frac{8}{19} x\ge -1$

Why did he multiply both sides by 19/8 and not 8/19 ? Is this a rule when dealing with inequalities that to remove fractions, you have to multiply by the reciprocal ?
8
votes
2answers
114 views

Prove $(1-a)(1-b)(1-c)(1-d)\geq abcd$ if $a^2+b^2+c^2+d^2=1$

Let $a,b,c,d\geq0$, $a^2+b^2+c^2+d^2=1$ Prove $\displaystyle (1-a)(1-b)(1-c)(1-d)\geq abcd$ I mutiplied both with $\displaystyle (1+a)(1+b)(1+c)(1+d)$ to use $1-a^2=b^2+c^2+d^2$ and try using the ...
2
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1answer
66 views

How prove $\frac{\sqrt{2}}{3}n^2<\sum_{k=1}^{n^2-1}\sqrt{1-\frac{\sqrt{k}}{n}}<\sqrt{2}n^2$

Show that $$\dfrac{\sqrt{2}}{3}n^2<\sqrt{1-\dfrac{\sqrt{1}}{n}}+\sqrt{1-\dfrac{\sqrt{2}}{n}}+\sqrt{1-\dfrac{\sqrt{3}}{n}}+\cdots+\sqrt{1-\dfrac{\sqrt{n^2-1}}{n}}<\sqrt{2}n^2.$$ Maybe use ...
15
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3answers
397 views

About an inequality including arithmetic mean, geometric mean and harmonic mean

For any $n$ positive real numbers $a_i\ (i=1,2,\cdots,n)$, let us define $A,G,H$ as $$A=\frac{\sum_{i=1}^{n}a_i}{n},\ G=\sqrt[n]{\prod_{i=1}^{n}a_i},\ H=\frac{n}{\sum_{i=1}^{n}\frac{1}{a_i}}.$$ ...
2
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0answers
77 views

Proving probability inequality (how to return to Chebychev?)

Supposing $X$ is a random variable, $X>0$, $E[X^2]<+\infty$, $\lambda \in (0,1)$, I have to prove the following inequality. $$P[X>\lambda E[X]] \geq (1-\lambda)^2 \frac{E[X]^2}{E[X^2]}$$ ...
3
votes
2answers
72 views

How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$?

I am reading a chapter about mathematical proofs. As an example there is: Prove that: $$(1) \space\space\space\space\space\space\space\space\space\space\space \frac{a+b}{2} \geq \sqrt{ab}$$ for ...
1
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1answer
25 views

Order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$

There is a multiple choices which syas what is the order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$? a. 1 b. 3 c. 2 d. empty I know that by considering certain cases, for example when $x<0$ or ...
0
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1answer
552 views

generalized inequalities defined by proper cones

The generalized inequality defined by a proper cone $K$ is that $x \ge_{K} y$ if $x-y \in K$ for $x,y \in K$. Does this means that for any $x \in K$, we have $x \ge_{K} 0$ since $x - 0 = x \in K$ ? ...
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0answers
18 views

Inequality involving special functions

Denote by ${}_2F_1$ the hypergeometric function (see https://en.wikipedia.org/wiki/Hypergeometric_function). Let $n,m\in\mathbb{Z}$ s.t. $2\leq m\leq n$, $$f_k\left(x\right) \equiv ...
6
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0answers
106 views
+100

How inequalities are made

I've been solving a lot of math contest inequality problems last few days and sometimes when I solve the problem I can easily ''see'' the idea behind it's creation (for an example, one clever ...
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0answers
15 views

Deriving an estimate in regularity theory of the heat equation

I have another question from PDE Evans 2nd edition, this time from pages 380-381. It's about a step in the formal derivation of estimates. Given the initial-value problem for the heat equation ...
3
votes
4answers
77 views

If $a,b,c$ are positive, then $(a+b+c)(1/a+1/b+1/c)\ge 9$

The question asks to prove that if "$x_1,x_2,x_3$ are positive numbers show that: $$(x_1+x_2+x_3) \left(\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3} \right)\ge 9$$ I've tried to use the fact that the ...
1
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1answer
59 views

Using integral estimation to show that $ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$

Show with Integral estimation that $$ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$$ $$f(k)=\frac {\ln k}{k^2} $$ For the integral it is : 1 But the other part is the ...
0
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1answer
27 views

Arithmetic and Geometric Mean Inequalities [on hold]

Can someone help me to understand the logic of: $$\sqrt{ab} \le \frac{a+b}{2}$$ Proof: ?
1
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4answers
67 views

If $x,y$ are positive, then $\frac1x+\frac1y\ge \frac4{x+y}$

For $x$, $y$ $\in R^+$, prove that $$\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}$$ Could someone please help me with this inequality problem? I have tried to use the AM-GM inequality but I must be doing ...
3
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2answers
80 views

Does my proof of $|x+y| \le |x| + |y|$ make sense? How do I conclude a proof?

Thank you for reading it. I know I made a lot of mistakes. This is my first ever proof that I have attempted. Another note is that I only have been studying proofs for about a week. Any advice will be ...
1
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1answer
32 views

An inner product inequality

In this article: http://rgmia.org/papers/v7e/RBKIIPS.pdf, the author claims that the inequality (after (2.4)) $$\frac{|\langle a,x\rangle \langle x,b\rangle|}{\|x\|^2} \leq ...
1
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1answer
44 views

Two sequences defined by recurrence relations satisfy $x_n/y_n<\sqrt{7}$ for all $n$

Let $(x_n)_{n\geq 1}$ and $(y_n)_{n\geq 1}$ be two sequences such that: $$x_{n+1}=x_n^2+1 \quad \text{ and } \quad y_{n+1}=x_n y_n$$ with $x_1=2$ and $y_1=1$ Prove that for all $n$ ...
0
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2answers
37 views

Largest number of pairs that can be added while keeping the population at least 60% male

I'm doing problems from the AoPS Algebra Beginner's book. There's this problem that states the following, At her ranch, Georgia starts an animal shelter to save dogs. After the first three days, she ...
0
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1answer
60 views

An extremal problem using AM-GM inequality

Let $x, y, z$ be nonnegative real numbers and such that $$ x^2+y^2+z^2=2. $$ Find the maximum value of $$ P=\frac{x^2}{x^2+yz+x+1}+\frac{y+z}{x+y+z+1}-\frac{1+yz}{9}. $$ My attempt. I guess that $P$ ...
0
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2answers
42 views

When does the equality hold for norm equivalence

We know that for a vector $x\in \mathbb{R}^n$, its 1-norm and 2-norm satisfy that $$\frac{1}{n}\|x\|_1\le\|x\|_2\le \|x\|_1,$$ could anyone please give me some hints that on what condition these ...
1
vote
2answers
87 views

Conditional extreme value of a function

Let $x,y,z$ be the positive real numbers, if $x^2+y^2+z^2=1$, then how can we find the minimal value of this function $f(x,y,z)=\dfrac{xz}{y}+\dfrac{yz}{x}+\dfrac{xy}{z}$.
0
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0answers
41 views

Can the inequality $a^3 + b^3 + c^3 \ge a^2b + ac^3 + b^2c$ be derived from arithmetic-geometric means? [duplicate]

The inequality goes as follow: $$a^3 + b^3 + c^3 \ge a^2b + ac^3 + b^2c$$ Where $a,b,$ and $c$ are positive real numbers. Also, can it be solved using am-gm?
1
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2answers
38 views

Upper bound on the entropy of a sum two random variables

Let $X$ be a random variable such that $|X| \leq A$ almost surely, for some $A > 0$. Let $Z$ be independent of $X$ such that $Z \sim {\cal N}(0, N)$. My question is: How large can the entropy ...
-4
votes
0answers
39 views

Calcul of limit [on hold]

What is the limit of $$\lim_{f \rightarrow 0} \frac{ \nabla {f(x)} }{\sin{(f(x))}}?$$ We can use the Poincare inequality and the famous limits: $$\lim_{x\rightarrow 0} ...
1
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1answer
43 views

Prove $a^2b+b^2c+c^2a \ge\sqrt{3(a^2+b^2+c^2)}$ if $abc=1$

if $a,b,c$ are positive real numbers that $abc=1 $,Prove:$$a^2b+b^2c+c^2a \ge\sqrt{3(a^2+b^2+c^2)}$$ Additional info: We should only use AM-GM and Cauchy inequalities. Things I have done so ...
2
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2answers
28 views

Taking root from absolute expression

Why is the following true? (Where all terms are positive) $$|x-y| < \epsilon^2 \implies |\sqrt x - \sqrt y| < \epsilon$$
0
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3answers
61 views

Why do the relations $ab=1/2$ and $a>b$ imply $a^2>1/2>b^2$ for positive $a,b$?

When I was reading a probstat book, I encountered an example which I am able to understand except for a formula which I am not able to grasp. It may be basic but I am not able to get it, the solution ...
4
votes
1answer
315 views

Stochastic integral inequality

Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$. Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that ...
3
votes
3answers
871 views

Simple high school inequality question

I'm reviewing some high school basic algebra and I would like to know when this function: $$2x-1-\frac{1}{x}$$ is positive. Solution Suppose $x\gt0$, then ...
3
votes
3answers
221 views

How to prove this inequality $\sum_{i=1}^{n}\left(x^k_{i}\ln{x_{i}}\ln{\frac{x_{i}}{n}}\right)\le 0$

Let $x_{i}\ge 0$ for $i\in\{1,2,\cdots,n\}$ and $x_{1}+x_{2}+\cdots+x_{n}=n$ for $n\ge 3$ Show that for all strictly positive integers $k\ge2$ the following inequality holds : ...
2
votes
3answers
101 views

How to prove this inequality without using Muirhead's inequality?

I ran into a following problem in The Cauchy-Schwarz Master Class: Let $x, y, z \geq 0$ and $xyz = 1$. Prove $x^2 + y^2 + z^2 \leq x^3 + y^3 + z^3$. The problem is contained in the chapter ...
5
votes
0answers
217 views

Inequality between incomplete beta and gamma functions

Let the regularized incomplete beta and gamma functions be defined as usual: \begin{equation} I_p(z,w) = \frac a {B(z,w)} \int_0^p t^{z-1} (1-t)^{w-1} \,\mathrm dt, \end{equation} \begin{equation} ...
0
votes
1answer
39 views

Find largest integer k that renders the inequality true.

$\frac{3}{2} \times \frac{2}{1} \times \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \ldots \times \frac{k}{k+1} \ge \frac{1}{8}$ What I've tried so far: $\frac{36}{48} \times \frac{k}{k+1} > ...
2
votes
1answer
79 views

Minimizing the expression $(1+1/x)(1+m/y)$ over positive reals such that $mx+y=1$

Let $x$ and $y$ be positive real numbers such that $mx+y=1$. Find the positive $m$ such that the minimum of: $$\left( 1 + \frac{1}{x} \right)\left( 1 + \frac{m}{y} \right).$$ is $81$. I have ...
2
votes
2answers
67 views

Show that $(n+1)^{n+1}>(n+2)^n$ for all positive integers

Show that: $(n+1)^{n+1}>(n+2)^n$ holds for all positive integers I tried using induction: for $n=1$ we have 4>3 then for $n+1$ we have to show that $(n+2)^{n+2}>(n+3)^{n+1}$ and here I ...
1
vote
1answer
23 views

Existence of a bound for sign functions?

Is the following statement true? $\forall a,b \in \mathbb{R}^d, ~ (\text{sgn}(a)^T\text{sgn}(b))^2\le (a^Tb)^2$ where $\text{sgn}(x) = \{1 ~\text{if} ~x \ge 0, -1 ~\text{if} ~x < 0 \}$.
3
votes
1answer
443 views

Application of Schwarz lemma

Each analytic function mapping the right half complex plane into itself must satisfy $$ \left|\frac{f(z)-f(1)}{f(z)+f(1)}\right| \leqslant \left|\frac {z-1}{z+1} \right|$$ for $\text{Re}\; z > 0.$ ...
4
votes
1answer
61 views

Connection between arithmetic mean, geometric mean and sample variance

Let $x_1, \dots, x_n$ be positive real numbers. Arithmetic-geometric mean inequality tells us that: $GM = \sqrt[n]{x_1 \dots x_n} \leq \frac{x_1 + \dots + x_n}{n} = AM$ and that equality occurs iff ...
2
votes
4answers
67 views

How to find the minimum value of $\sum_{1\le i<j\le 6}[a_{i}+a_{j}]$

let $a_{1},a_{2},\cdots,a_{6}$ be real numbers,and such $$a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}=2014$$ Find the minimum of the value $$\sum_{1\le i<j\le 6}[a_{i}+a_{j}]$$ where $[x]$ is the ...
3
votes
2answers
128 views

Prove $\left(\sum_{1 \le i <j \le n} |x_i-x_j|\right)^2 \ge (n-1)\sum_{1\le i<j \le n} (x_i-x_j)^2.$

Question: Let $x_1,\, x_2,\,\ldots,\, x_n$ be real numbers. Prove that $$\left(\sum_{1 \le i <j \le n} |x_i-x_j|\right)^2 \ge (n-1)\sum_{1\le i<j \le n} (x_i-x_j)^2.$$ This problem is ...
6
votes
3answers
1k views

Why is this true: $\|x\|_1 \le \sqrt n \cdot \|x\|_2$?

Stuck, help, please, I am really new to this. I opened the 2-norm and multiplied by $n$, then I am thinking to square both sides. The problem is that I do not know how to open $(x_1 + x_2 + ... + ...