Questions on proving and manipulating inequalities.

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2
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0answers
12 views

Inequalities with $\|x-y\|$, $|\langle x,y\rangle|$, and $\sqrt{\|x\|^{2}+\|y\|^{2}}$ in a Hilbert space

Let $H$ be a Hilbert space, and let $\|x\|$ denote the norm of $x\in H$, and $\langle x,y\rangle$ denote the inner product of $x,y\in H$. For $x,y\in H$ let us denote $\alpha(x,y)=\|x-y\|$, ...
0
votes
1answer
15 views

Inequality involving products

One is given two intervals $I_{a-\epsilon,b+\epsilon}$, $I_{a,b}$ of $\mathbb{R}^n$, and is asked to show that $\lambda(I_{a-\epsilon,b+\epsilon}) - \lambda(I_{a,b}) \leq c\epsilon$ for some constant ...
1
vote
0answers
32 views

A basic inequality problem

$$(\frac{\pi}{22}) \cos (\frac{\pi}{22}) +(\frac{2\pi}{11}) \cos (\frac{5\pi }{22}) + (\frac{2\pi}{ 11}) \cos (\frac{9\pi}{22}) + (\frac{\pi}{22}) \cos(\frac{5\pi}{11}) < (\frac{\pi}{26}) ...
0
votes
0answers
16 views

Minimum of summed sequence

Define M non-negative sequences, \begin{equation} a_{m,1}\geq a_{m,2}\geq,...,\geq a_{m,K}\quad \text{for}\ m=1,..,M \end{equation} and cyclic shifted versions $a^{\zeta_m}_{m,k}$ with shift value ...
0
votes
0answers
31 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 ...
0
votes
0answers
27 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.
1
vote
3answers
42 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 ...
2
votes
4answers
54 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 ...
0
votes
0answers
54 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) ...
4
votes
6answers
439 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 ?
3
votes
2answers
73 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
vote
1answer
34 views

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

There is a multiple choices which says 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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
0answers
16 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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
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: ?
3
votes
4answers
78 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
vote
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 ...
3
votes
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 ...
0
votes
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 ...
1
vote
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
votes
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 ...
0
votes
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
vote
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
votes
2answers
28 views

Taking root from absolute expression

Why is the following true? (Where all terms are positive) $$|x-y| < \epsilon^2 \implies |\sqrt x - \sqrt y| < \epsilon$$
-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} ...
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 ...
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} > ...
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 $\text{sgn}(x) = \{1 ~\text{if} ~x \ge 0, -1 ~\text{if} ~x < 0 \}$.
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 ...
2
votes
0answers
80 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. $$P[X>\lambda E[X]] \geq (1-\lambda)^2 \frac{E[X]^2}{E[X^2]}$$ ...
6
votes
0answers
107 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 ...
4
votes
1answer
62 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 ...
1
vote
2answers
34 views

a proof of constants are null from a given inequality

Problem: given constants $a,b\text{ and }c$, and a variable $x$, assume that for all $x\in\mathbb{R}$ holds that $|ax^2+bx+c|\le|x|^3$, then proof that $a=b=c=0$ My try: substitute $x=0$ into the ...
0
votes
1answer
15 views

Nesbitt inequality symmetric proof

I was trying to proof Nesbitt inequality using symmetry. So i asummed that $$a+b=x , b+c =y ,c+a=z$$ and then I rewrite inequality as:$$\sum \limits_{cyc} \frac {\frac {x+z-y}{2}}{y}\geq ...
3
votes
1answer
77 views

Prove $a^3+b^3+c^3\geq a^2b+b^2c+c^2a$ [duplicate]

if $a,b,c$ are positive real numbers,Prove:$$a^3+b^3+c^3\geq a^2b+b^2c+c^2a$$ Things I have done so far: I know the fact that $$a^3+b^3+c^3\geq\frac{1}{2}[ab(a+b)+bc(b+c)+ca(c+a)]$$ However i ...
0
votes
1answer
126 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 ...
1
vote
0answers
52 views

Comparing a number with a line of power

How do you compare which is bigger (or maybe equal), LHS or RHS, in $$a \sim b_1^{b_2^{.^{.^{.^{b_n}}}}}$$ given $a$ and $b_i$, $1 \leq i \leq n$, are non-negative integers (also could be big)? The ...
0
votes
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 ...
0
votes
1answer
24 views

Applying Markov's inequality to a sequence of random variables

Does the Markov inequality also work for infinite $a$ or only for constant $a$? More precisely: If $X(n)$ is a sequence of random variables and $f(n)$ is some sequence of numbers,is it allowed to ...
2
votes
2answers
69 views

Proving that a number is non-negative?

The numbers $a$,$b$ and $c$ are real. Prove that at least one of the three numbers $$(a+b+c)^2 -9bc \hspace{1cm} (a+b+c)^2 -9ca \hspace{1cm} (a+b+c)^2-9ab$$ is non-negative. Any hints would be ...
1
vote
0answers
91 views

How to prove this inequality $b(a-1)(c-1)+c(b-1)(a-1)+a(c-1)(b-1)\le 0$

let $a,b,c>0$,and $$abc=1$$ show that $$b(a-1)(c-1)+c(b-1)(a-1)+a(c-1)(b-1)\le 0$$ since $$b(a-1)(c-1)=b(ac-a-c+1)=abc-ab-bc+b=1-ab-bc+b$$ so we only prove $$3-2(ab+bc+ac)+a+b+c\le 0 $$ oh,this ...
0
votes
1answer
24 views

Continued fraction inequality: $q_n\left|q_n\alpha-p_n\right|(a_{n+1}+1)>1$

In an article it is used the fact that $$q_n\left|q_n\alpha-p_n\right|(a_{n+1}+1)>1$$ where $\alpha=[a_0;a_1,\ldots]$ is an irrational number and $q_i$ is the series of the best approximation ...
4
votes
2answers
56 views

Find the smallest constant K satisfying the inequality

Find the smallest constant $K$satisfying the inequality $$x^{1\over 3}+y^{1\over 3} \le K(x+y)^{1\over 3}$$ The official proof makes the substitution $a=x^{1\over 3}$ and $b=y^{1\over 3}$, which does ...
0
votes
1answer
28 views

Alternative solutions of an inequality problem

Let $x, y, z$ be distinct real numbers. Prove $ \sqrt[3]{x - y} + \sqrt[3]{y - z} + \sqrt[3]{z - x} \neq 0$ I'm curious about different ways to solve this inequality. My solution:
1
vote
2answers
42 views

How to show that $\frac {q + \frac {1}{2}}{p - \frac {1}{2}} > \sum_{i = p}^q\frac {1}{i}$ if $q\ge p > 0?$

How to show that : $$\frac{2q+1}{2p-1}>\sum_{i=p}^q\frac{1}{i}$$ if $q\ge p>0$
1
vote
2answers
54 views

Prove Schwarz inequality in $R^2$

Can someone please show me how you would prove the following in $R^2$ $\int f(x)* g(x) dx \leqslant \int f(x)^2 dx * \int g(x)^2 dx $ starting from $\int [\lambda*f(x) - g(x)]^2 dx \geqslant ...