Questions on proving and manipulating inequalities.

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4
votes
2answers
36 views

How two divide two inequalities

I would like to know if somebody knows how to properly divide one inequality by another, as a resolution method similar to when we divide one equality by another. Take this as an example: $x^2 - y^2 ...
0
votes
1answer
42 views

Symmetry in inequalities.

I previously asked a question Is Symmetry a valid option in Inequalities. After some thinking I concluded something: Some theorem which I made up:[Call it my theorem :D] Let $f$ be a ...
0
votes
2answers
26 views

Can we show that $E\|X-Y\|^2 \leq E\|X-Z\|^2 + E\|Z-Y\|^2$

Let $X,Y,Z$ be some random elements on some Hilbert space $(H,\langle\cdot,\cdot\rangle)$. Can we show that $$E\|X-Y\|^2 \leq E\|X-Z\|^2 + E\|Z-Y\|^2$$ I can clearly see that $$E\|X-Y\|^2 \leq ...
2
votes
3answers
68 views

Prove $3(x^2y+y^2z+z^2x)(xy^2+yz^2+zx^2)\ge xyz(x+y+z)^3$

if $x,y,z$ are positive real numbers,Prove:$$3(x^2y+y^2z+z^2x)(xy^2+yz^2+zx^2)\ge xyz(x+y+z)^3$$ Additional info:$\sum_{cyc}$ denotes sums over cyclic permutations of the symbols $x,y,z$. I'm ...
1
vote
2answers
31 views

Prove or disprove $xyz+\frac{8}{27}\ge xy+yz+zx$ if $x+y+z=1$

if $x,y,z$ are positive and $x+y+z = 1$,Prove:$$xyz+\frac{8}{27}\ge xy+yz+zx$$ Additional info:I'm looking for solutions and hint that using Cauchy-Schwartz and AM-GM because I have background ...
0
votes
1answer
25 views

How to get this inequality

Let $c>0$, $n \in \mathbb N$ and $q>1$. How to get the following approximating inequality when $n$ is large, please? To be more specific, I cannot see how to get rid of the square root. $$ ...
1
vote
1answer
35 views

Prove $(x+y)(y+z)(z+x)\ge\frac{8}{3}(x+y+z)\sqrt[3]{x^2y^2z^2}$

if $x,y,z$ are positive real numbers,Prove:$$(x+y)(y+z)(z+x)\ge\frac{8}{3}(x+y+z)\sqrt[3]{x^2y^2z^2}$$ Additional info:I'm looking for solutions and hint that using Cauchy-Schwartz and AM-GM ...
1
vote
2answers
36 views

Prove $\sum_{cyc} \frac{\sqrt{xy}}{\sqrt{xy+z}}\le\frac{3}{2}$ if $x+y+z=1$

if $x,y,z$ are positive real numbers and $x+y+z=1$ Prove:$$\sum_{cyc} \frac{\sqrt{xy}}{\sqrt{xy+z}}\le\frac{3}{2}$$ where $\sum_{cyc}$ denotes sums over cyclic permutations of the symbols ...
-2
votes
0answers
22 views

Riemann's sum inequality problem [on hold]

Iam having touble with a certain question on my assignment. I dont know how to replicate the math symbols on this site so I have jst put down a link to the full assignment: ...
1
vote
2answers
33 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
votes
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
votes
1answer
21 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: ...
-1
votes
0answers
20 views

on CS inequality [on hold]

The CS inequality is generalisation of triangle inequality . Is it true. I think it is TRUE . but I am not sure. Please explain. and help me to solve this let m,n,x,y,z be +ve real numbers with ...
2
votes
2answers
86 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
votes
3answers
41 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| ...
-1
votes
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)} ...
0
votes
3answers
42 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.
1
vote
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
votes
0answers
11 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
vote
1answer
57 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).
1
vote
0answers
17 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
votes
5answers
58 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 $\sin A+\sin B+\sin C$
0
votes
2answers
38 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 ...
-1
votes
3answers
45 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
70 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! ...
4
votes
2answers
68 views

An uncanny inequality with Gamma function [on hold]

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.
1
vote
0answers
31 views

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

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
votes
2answers
105 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 ...
2
votes
1answer
125 views
+50

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

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
32 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 ...
0
votes
0answers
19 views

General Form Radical Inequality

$±\sqrt{P^n (x)} ± \sqrt{Q^n (x)} < ± \sqrt{R^n (x)} ± \sqrt{S^n (x)}$ I am having a tedious time proving this inequality for the four polynomials of the nth degree. I have squared each side at ...
4
votes
1answer
62 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
votes
4answers
137 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
votes
0answers
73 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
votes
3answers
79 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
154 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
votes
1answer
33 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
vote
0answers
62 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
votes
0answers
24 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
37 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
28 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
54 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
64 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
55 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
455 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
91 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
52 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 ...
1
vote
0answers
87 views
+300

Inequality involving Pochhammer symbols

I am trying to show that $$ ...