Questions on proving, manipulating and applying inequalities.

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1
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2answers
92 views

prove that $ a^2 + b^2 + c^2 \ge 2\left( {a^3 b^3 + a^3 c^3 + c^3 b^3 + 4a^2 b^2 c^2 } \right)$

I have: let $$a;b;c$$ be non-negative real numbres with sum 2.prove that $$a^2 + b^2 + c^2 \ge 2\left( {a^3 b^3 + a^3 c^3 + c^3 b^3 + 4a^2 b^2 c^2 } \right)$$ I should determine whether this ...
5
votes
1answer
926 views

Help with problem 1-5 from Calculus by Spivak (multiplying two inequalities together)

This question involves problem 5(viii) from chapter 1 of Spivak's Calculus, third edition. I'm a layman trying to teach myself some more advanced mathematics, and although I've been making slow ...
3
votes
1answer
67 views

How prove $\sqrt{x}+\sqrt{y}+\sqrt{\frac{x+y+2}{xy-1}}\ge 2\left( \frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}+\sqrt{\frac{xy-1}{x+y+2}}\right)$

How prove $\sqrt{x}+\sqrt{y}+\sqrt{\frac{x+y+2}{xy-1}}\ge 2\left( \frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}+\sqrt{\frac{xy-1}{x+y+2}}\right)$ for $8x\ge13, 8y \ge 13$?
1
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2answers
115 views

$(x+y)^p\leq x^p+y^p$ for positive x, y and 0<p<1 [duplicate]

I need to use an inequality $(x+y)^p\leq x^p+y^p$ for positive $x$, $y$ and $0<p<1$ in a proof, but I'm not sure how to derive it and I wasn't able to find it anywhere. I think it might hold, ...
1
vote
3answers
126 views

Considering a convex polygon lying on a plane in 3D space, how can I know if a point on that plane lies inside or outside that polygon?

I have a plane in space and a polygon in it. I know the position of each vertices making the polygon. I also know the position of the point on the plane. How can I know whether the point is inside or ...
2
votes
1answer
200 views

How prove this inequality $a^3b+b^3c+c^3a+a^3b^3+b^3c^3+c^3a^3\le 6$

let $a,b,c>0$, and such $$a^2+b^2+c^2=3$$ show that $$a^3b+b^3c+c^3a+a^3b^3+b^3c^3+c^3a^3\le 6\tag{2}$$ I know this famous inequality( creat by valsie) $$(a^2+b^2+c^2)^2\ge 3(a^3b+b^3c+c^3a)$$ ...
-2
votes
1answer
33 views

Can one apply absolute value to both sides of an inequality?

Can I claim this: $$c>b \implies |c|>|b|$$ with $c>0$? I ask because I want to use it in a proof but I am not sure. Thank You.
2
votes
2answers
90 views

Show that $x+y\geq x^\alpha y^{1-\alpha}$

Is it possible to show that $$x+y\geq x^\alpha y^{1-\alpha}$$ for $\alpha \in(0,1]$ and $x,y\in[0,\infty)$? I tried to manipulate it algebraically, but it does not give me any anything. Equivalently, ...
3
votes
3answers
445 views

Wolfram|Alpha returns the wrong result: how can I solve this “high precision” equation?

$$1-(1-1.40*10^{-36})^x \ge 1.09*10^{-9}$$ I want to estimate $x$ such that the probability on the left becomes larger than the probability on the right. A solution must exist because ...
4
votes
5answers
89 views

How find the minimum $\frac{(5y+2)(2z+5)(x+3y)(3x+z)}{xyz}$,if $x,y,z>0$

let $x,y,z>0$, find the minimum of the value $$\dfrac{(5y+2)(2z+5)(x+3y)(3x+z)}{xyz}$$ I think we can use AM-GM inequality to find it. $$5y+2=y+y+y+y+y+1+1\ge 7\sqrt[7]{y^5}$$ ...
2
votes
2answers
117 views

Any two points inside a circle are within a diameter of each other.

In many problems involving the Pigeonhole Principle, we often assume the following lemma: Lemma: The distance between any two points in a circle of radius $r$ is at most $2r$. Intuitively, this ...
7
votes
1answer
64 views

Linear inequality problem: $2x + 1 > 10$

$2x + 1 > 10$ $2x > 9$ $x > 4.5$ The answer in the book says: $x\lt 4.5$. Am I doing it wrong?
5
votes
2answers
83 views

Lower bound for $2\sin^2(y\pi)$

I was trying to understand the proof of a theorem, and the author uses the fact that if $y \in \mathbb{Q} \cap(0, \frac{1}{2}]$, then $$2\sin^2(y\pi) \geq \frac{8}{n^2},$$ where $y=\frac{p}{q}$, ...
1
vote
1answer
423 views

Inequalities for the probability of union and intersection of events

Prove that $\min(1, P(A)+P(B))\ge P(A\cup B)$ $\min(P(A),P(B))\ge P(A\cap B)\ge \max(0,P(A)+P(B)-1)$ Where $A$ and $B$ are events. I don't know how to prove them; Can you give me a hand please?, ...
3
votes
1answer
67 views

Matrix inequalities question

Let $A, B \in \mathbb{R}^{n \times n}$. Assume that: $$ 0 \preccurlyeq 2 A^\top A \preccurlyeq A^\top + A $$ $$ B^\top + B \preccurlyeq 0 $$ Is the following inequality true? $$ A B + B^\top ...
3
votes
1answer
84 views

Any nice way to find number number of single digit ordered pairs $(a, b)$ such that $a!b! \gt a!+b!$

I have listed them all by brute force : a = 0,1 : no solutions a = 2 : b = 3,4,5,...9 c = 3 : b = 2,3,4...9 I'm wondering if there is a clever approach to ...
4
votes
2answers
111 views

Prove $\frac{ab}{1+c^2}+\frac{bc}{1+a^2}+\frac{ca}{1+b^2}\le\frac{3}{4}$ if $a^2+b^2+c^2=1$

Ff $a,b,c$ are positive real numbers that $a^2+b^2+c^2=1$ ,Prove: $$\frac{ab}{1+c^2}+\frac{bc}{1+a^2}+\frac{ca}{1+b^2}\le\frac{3}{4}$$ Additional info: I'm looking for solutions and hint that ...
6
votes
2answers
239 views

How prove this $\prod_{i=1}^{r}\left(1+\frac{1}{x_{i}}\right)\le \frac{2^{2^r}-1}{2^{2^r-1}}$

Let $x_{1},x_{2},\cdots,x_{r}$ be positive integers such that $$1\le x_{1}\le x_{2}\le \cdots\le x_{r}$$ and $$\prod_{i=1}^{r}\left(1+\dfrac{1}{x_{i}}\right)<2.$$ then Show that ...
3
votes
2answers
84 views

Prove $\frac{a}{1+b^2}+\frac{b}{1+c^2}+\frac{c}{1+d^2}+\frac{d}{1+a^2} \ge 2$ if $a+b+c+d=4$

if $a,b,c,d$ are positive real numbers that $a+b+c+d=4$,Prove:$$\frac{a}{1+b^2}+\frac{b}{1+c^2}+\frac{c}{1+d^2}+\frac{d}{1+a^2} \ge 2$$ Additional info:I'm looking for solutions and hint that ...
3
votes
2answers
153 views

How to prove this $\frac{\sin{(nx)}}{\sin{x}}\ge\frac{\sqrt{3}}{3}(2n-1)^{\frac{3}{4}}$

let $n<\dfrac{\pi}{2\arccos{\dfrac{c}{2}}},c\in (0,2),c=2\cos{x}$, show that $$\dfrac{\sin{(nx)}}{\sin{x}}\ge\dfrac{\sqrt{3}}{3}(2n-1)^{\frac{3}{4}}$$ where $0<x<\dfrac{\pi}{2}$ My idea: ...
-1
votes
3answers
78 views

Triangle Inequality on complex numbers

Problem Let $z= x + iy$, then prove that: $$|x| + |y| \le 2 ^{1/2} |z|$$ Progress I've tried to write $|z|$ as $(x^2 + y^2)^{1/2}$, and to make some algebra after this, but I'm really new at ...
0
votes
1answer
52 views

Ordinal multiplication property: $\alpha<\beta$ implies $\alpha\gamma\le\beta\gamma$

I'm having trouble proving the following two ordinal multiplication properties. If $\alpha, \beta$, and $\gamma$ are such that $\alpha \lt \beta$ and $\gamma \gt 0$, then $\alpha\gamma \le ...
1
vote
2answers
511 views

Comparison theorem for ODE

Here is something I'm trying to prove: Conjecture: Suppose $f'(x) \leq \phi(f(x), x)$ and $f(a)=\alpha$. Suppose $g'(x)=\phi(g(x),x)$ and $g(a)\geq \alpha$. Then $f(x)\leq g(x)\,\,\forall x$. ...
2
votes
1answer
50 views

Check correct delta in eps-delta proof

I been stuck now with this seemingly simple exercise for some time. I need to show that: $|x^2-4| < \epsilon$ when $0 < |x-2| < \epsilon(5+\epsilon)^{-1}$ But I'm at a loss. I know that I ...
1
vote
2answers
144 views

Tricky geometry proof

If a,b,c belong to the interval $(0,1)$ and $ab + ca + bc = 1$, prove that $$\frac{a}{1-a^2} + \frac{b}{1-b^2} + \frac{c}{1-c^2}\ge\frac{3^{3/2}}{2}$$ How would you go about solving such a problem?
2
votes
2answers
99 views

Ordinal numbers addition property: $b<c$ implies $b+a \le c+a$

I'm having trouble proving the following property of ordinal numbers. If $a, b, c$ are ordinal numbers such that $b \lt c$, then $b+a \le c+a$. I first started by assuming $g$ as an order ...
1
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1answer
33 views

How to get $x$ from $x(\ln(a/x))<c$?

Given an equality: $$x(\ln(a/x))<c$$ Where $a,c,x\in \mathbb N$ and $ a>x$, I'd like to find something of the form $$x > f(a,c)$$ It doesn't have to be the exact domain of $x$ which ...
2
votes
1answer
80 views

Proving Example 1.1.15 of secrets in inequalities

if $a,b,c,d$ are positive real numbers,Prove:$$\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)^2\ge\frac{1}{a^2}+\frac{4}{a^2+b^2}+\frac{9}{a^2+b^2+c^2}+\frac{16}{a^2+b^2+c^2+d^2}$$ ...
0
votes
2answers
71 views

Prove $n^n\prod_{i=1}^{n}(x_i^{n}+1) \ge \sum_{i=1}^{n} x_i +\sum_{i=1}^{n}\frac{1}{x_i}$

if $x_i$ is positive real number that $\prod_{i=1}^{n} x_i=1$,Prove:$$n^n\prod_{i=1}^{n}(x_i^{n}+1) \ge \sum_{i=1}^{n} x_i +\sum_{i=1}^{n}\frac{1}{x_i}$$ Additional info:I'm looking for solutions ...
4
votes
2answers
327 views

How to 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
2answers
231 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 ...
1
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2answers
114 views

Induction Proof of trig inequality $\sum_{k=0}^n |\cos k| \ge \frac n2$

This is for a course, I don't want the answer just a prod in the right direction! I've got a problem that states let n be an integer such that $$n\gt0$$ $$\text{Prove: }\sum_{k=0}^n |\cos k| \ge ...
1
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2answers
34 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
120 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
71 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-Schwarz and AM-GM because I have background ...
0
votes
1answer
45 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
89 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-Schwarz and AM-GM ...
2
votes
2answers
104 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 ...
1
vote
2answers
60 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-Schwarz and AM-GM because ...
0
votes
3answers
79 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
105 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
122 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
59 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| ...
0
votes
3answers
186 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
43 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 ...
1
vote
1answer
78 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
55 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
102 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
50 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
56 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