Questions on proving and manipulating inequalities.

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2answers
37 views

Prove $a^3+b^3+c^3\ge a^2+b^2+c^2$ if $ab+bc+ca\le 3abc$

if $a,b,c$ are positive real numbers and $ab+bc+ca\le 3abc$ Prove:$$a^3+b^3+c^3\ge a^2+b^2+c^2$$ Additional info:I'm looking for solutions and hint that using Cauchy-Schwartz and AM-GM because ...
0
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2answers
40 views

How to solve exponential inequality with $x$

I need to solve the following inequality. $$\ln(x) - x > 0.$$ I oddly remember that it can only be done by using the graph... Is it true? I have the same problem with $$e^x(x-1)>-2.$$ ...
0
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1answer
34 views

Expressing a solution in interval notation

I am faced with this problem. I am told to express the answer in interval notation. |3x| > 12 I solve like usual, by doing this: ...
2
votes
2answers
92 views

How to show that an infinite decimal is equal to a unique real number?

I don't understand how the proof above shows that two distinct real numbers correspond to different infinite decimal. All I got out of the explanation is given any two distinct real numbers $a$ and ...
0
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3answers
53 views

How to conclude $|a|<|b|$ from $a<\frac{b^2}{a} \text{ and } \frac{a^2}{b}<b$? (Direct Proof)

The original question is to prove that for all real numbers $a$ and $b$, $a^2 < b^2 \Rightarrow |a| < |b|$. I was able to easily prove this by proving that its contrapositive, $|a|\ge|b| ...
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0answers
14 views

$\big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\big\|_{H^1(\mathbb R)} \le C_{>0}\|f(x)\|_{L^\infty(\mathbb R)}^2 \|f(x)\|_{H^1(\mathbb R)}$ holds? [duplicate]

I want to know that whether the following inequality holds or not for complex-valued functions $f_1$, $f_2$, $f_3$ on $\mathbb R$: $$ \big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\big\|_{H^1(\mathbb R)} ...
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3answers
44 views

Do you flip the inequality sign if multiplying a quadratic equation by $-1$?

$$(-1)(-x^{ 2 }+3x+18)<0(-1)$$ becomes $$x^{ 2 }-3x-18>0\quad ?$$ I want to confirm before proceeding in solving a quadratic inequality.
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0answers
42 views

Inequality $\Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le C\|f(x)\|_{L^\infty(\mathbb R)}^2 \|f(x)\|_{H^1(\mathbb R)}$

For complex-valued functions $f_1, f_2, f_3:\mathbb R\to\mathbb C$, I want to know that the following inequality holds: $$ \Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le ...
0
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0answers
13 views

Existence for a differential inequality with mixed boundary conditions

For $\pi < \theta < 2\pi$, I have the following second-order differential inequality $$y''(s) (1 - \cos s - \tan \frac{\theta}{2} \sin s) + 2y'(s) (\sin s - \tan \frac{\theta}{2} \cos s) + y(s) ...
1
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1answer
63 views

Prove $1! + 2! + . . . + n! < (n + 1)!$ using mathematical induction [duplicate]

$1! + 2! + . . . + n! < (n + 1)!$ This question has left me stumped for quite some time. I am not sure how to approach it. (I am really bad at induction).
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0answers
19 views

Bounding the norm of the product of random PSD matrices

Consider the following setup, $(X, \hat{X}, Y, \hat{Y})$ are four $n \times n$ real, symmetric, full-rank, positive-definite matrices with entries between zero and one and operator norm $O(n)$. The ...
2
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5answers
62 views

To find maximum value

If $A>0,B>0$ and $C>0$ and further it is known that $A+B+C=\frac{5\pi}{4}$,then find the maximum value of $\sin A+\sin B+\sin C$
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2answers
40 views

Prove $\left | \sum_{k=1}^{n} a_k \right | \leq\sum_{k=1}^{n} \left | a_k \right |$

Prove that if $a_1,...,a_n$ are real numbers then $\left | \sum_{k=1}^{n} a_k \right | \leq \sum_{k=1}^{n} \left | a_k \right |$ Where $|x|$ is the absolute of $x$ I dont have any idea, how to ...
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3answers
48 views

how to prove $1/n (1-(1/2)^n)$ decreasing without using differentiation

$a(n)=1/n (1-(1/2)^n)$ prove $a(n+1)<a(n)$ for n>0 by differentiating slope comes negative and then we can prove it . but i wanted to solve it without that . can someone help
2
votes
2answers
78 views

How to prove $n! > n^a$ for all $a\in \mathbb{R}$ (for sufficiently large $n$)?

I've encountered a proof which claims $n! > n^2$ for sufficiently large $n$. I tried using induction to prove it for an arbitrary $a$: $n! > n^a$. Lets assume the claim is true for $n$: $n! ...
5
votes
2answers
74 views

An uncanny inequality with Gamma function [closed]

Prove for $x>0$ that $$ \frac{\Gamma^{\prime}(x+1)}{\Gamma(x+1)}>\log x$$ How to prove this inequality? thanks. This is a problem from Miklos Schweitzer Competition.
0
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0answers
34 views

Why $\alpha\le L_1$ for any $L_1>L$ implies $\alpha\le L$? [closed]

First part Second part Let $\alpha=\limsup|s_n|^{1/n}$ and $L=\limsup\left|\frac{s_{n+1}}{s_n}\right|$. We need to prove $\alpha\le L$. This is obvious if $L=+\infty$, so we assume $L<+\infty$. ...
2
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1answer
114 views

Use induction to prove the following: $1! + 2! + … + n! \le (n + 1)!$

Use induction to prove the following: $1! + 2! + .... + n! < (n + 1)!$ Base case: $n = 1$ $1! < 2!$ true Inductive step: Assume that $1! + 2! + .... + k! \le (k + 1)!$ is true let $n = k ...
5
votes
3answers
413 views

How prove this inequality $\sum\limits_{cyc}\frac{1}{a+3}-\sum\limits_{cyc}\frac{1}{a+b+c+1}\ge 0$

show that: $$\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}-\left(\dfrac{1}{a+b+c+1}+\dfrac{1}{b+c+d+1}+\dfrac{1}{c+d+a+1}+\dfrac{1}{d+a+b+1}\right)\ge 0$$ where $abcd=1,a,b,c,d>0$ I ...
3
votes
1answer
37 views

Prove logarithmic inequality with greatest integer function.

$\left \lfloor n\log_2 n^2 \right \rfloor + \left \lfloor \log_2(\left \lfloor n\log_2n^2 \right \rfloor) \right \rfloor \leq \left \lfloor (n+1)\log_2 (n+1)^2 \right \rfloor + 1$ How to show this? I ...
4
votes
1answer
66 views

Prove this inequality: $\sum{\frac{1}{(x+2y)^2}} \geq\frac{1}{xy+yz+zx}$

Let $x,y,z>0$: Prove that: $\frac{1}{(x+2y)^2}+\frac{1}{(y+2z)^2}+\frac{1}{(z+2x)^2} \geq\frac{1}{xy+yz+zx}$ I tried to apply Cauchy - Schwarz's inequality but I couldn't solve this solution!
6
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4answers
143 views

How prove this inequality $(1+\frac{1}{16})^{16}<\frac{8}{3}$

show that $$(1+\dfrac{1}{16})^{16}<\dfrac{8}{3}$$ it's well know that $$(1+\dfrac{1}{n})^n<e$$ so $$(1+\dfrac{1}{16})^{16}<e$$ But I found this $e=2.718>\dfrac{8}{3}=2.6666\cdots$ ...
0
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0answers
86 views

How prove this inequality with $\sqrt{a_{1}+\sqrt[3]{a_{2}+\sqrt[4]{a_{3}+\cdots+\sqrt[n+1]{a_{n}}}}}$

let $a_{i}>0,i=1,2,\cdots,n$, show that $$\sqrt{a_{1}+\sqrt[3]{a_{2}+\sqrt[4]{a_{3}+\cdots+\sqrt[n+1]{a_{n}}}}}\ge \sqrt[\displaystyle{(2!+3!+4!+\cdots+(n+1)!)}]{a_{1}a_{2}\cdots ...
2
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3answers
82 views

How find this $\frac{1}{x-y}+\frac{1}{y-z}+\frac{1}{x-z}$ minimum of the value

let $x,y,z\in R$,and such $x>y>z$,and such $$(x-y)(y-z)(x-z)=16$$ find this follow minimum of the value $$I=\dfrac{1}{x-y}+\dfrac{1}{y-z}+\dfrac{1}{x-z}$$ My idea: since ...
4
votes
3answers
160 views

Geometric interpretation of an integral inequality

Let $f: [a, b] \to \mathbb [0, \infty)$ be an integrable function. By Cauchy-Schwartz: $$ \left(\int_a^b f(x) dx\right)^2 \leq (b-a) \int_a^b f(x)^2 dx$$ with equality iff $f$ is constant. If we ...
3
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1answer
49 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
22 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
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0answers
79 views

Proving $\frac\pi{22}\cos\frac\pi{22}+\frac{2\pi}{11}\cos\frac{5\pi }{22}+\frac{2\pi}{ 11}\cos\frac{9\pi}{22}+\frac\pi{22}\cos\frac{5\pi}{11}<\cdots$

$$(\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
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0answers
27 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
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0answers
39 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 ...
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0answers
29 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
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3answers
59 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
66 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
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0answers
57 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) ...
3
votes
6answers
458 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
3answers
100 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
65 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
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1answer
67 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 ...
10
votes
2answers
501 views

Inequality involving Pochhammer symbols

Let $m,S$ be integers satisfying $2\leq m\leq S$. I would like to show that $$h_1\left(x\right) h_3\left(x\right) \leq h_2^2\left(x\right)$$ for all $x\geq 0$ where $$h_k\left(x\right) \equiv ...
2
votes
1answer
46 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
61 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
90 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
29 views

Arithmetic and Geometric Mean Inequalities [closed]

Can someone help me to understand the logic of: $$\sqrt{ab} \le \frac{a+b}{2}$$ Proof: ?
3
votes
4answers
91 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
40 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
97 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
38 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
48 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
51 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
43 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?