Questions on proving and manipulating inequalities.

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3
votes
1answer
39 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 ...
2
votes
2answers
66 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, ...
9
votes
4answers
248 views

Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$

Applying the Copson's inequality, I found: $$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where $\Psi^{(1)}(k)$ is the polygamma function. Is it know any ...
7
votes
1answer
56 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
267 views
+50

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
5answers
62 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. $$5y+2=y+y+y+y+y+1+1\ge 7\sqrt[7]{y^5}$$ ...
1
vote
0answers
86 views

Is my proof wrong?

Previously on an answer this, I received many downvotes, So I assume there's something wrong with it and no one did explain the reason, So I am asking a Question to it,I want if someone can tell me if ...
3
votes
3answers
369 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 ...
0
votes
0answers
23 views

Properties of Sup, Inf on sets.

I understand this proof is a bit long-winded, but I am only concerned with it is correct or not. It seems sound to me. Claim: If $A,B \subset \mathbb{R}$ and are non-empty, $a \leq b, \forall a \in ...
0
votes
0answers
21 views

Power series with complex variables inequality

I am struggling to prove the following inequality: For $z \in \mathbb{C}, r \in \mathbb{R}, n \in \mathbb{N}$, if $|z| \leq r$ and $1 \leq r < n$ then ...
2
votes
2answers
47 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 ...
3
votes
0answers
40 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 postive integer $x_{1},x_{2},\cdots,x_{r}$ such $$1\le x_{1}\le x_{2}\le \cdots\le x_{r}$$ and $$\prod_{i=1}^{r}\left(1+\dfrac{1}{x_{i}}\right)<2$$ show that ...
3
votes
2answers
116 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: ...
4
votes
2answers
60 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 $y=\frac{p}{q}$, ...
4
votes
2answers
66 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 ...
7
votes
2answers
672 views

Prove $\limsup\limits_{n \to \infty} (a_n+b_n) \le \limsup\limits_{n \to \infty} a_n + \limsup\limits_{n \to \infty} b_n$

I am stuck with the following problem. Prove that $$\limsup_{n \to \infty} (a_n+b_n) \le \limsup_{n \to \infty} a_n + \limsup_{n \to \infty} b_n$$ I was thinking of using the triangle inequality ...
0
votes
1answer
26 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 ...
0
votes
1answer
27 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?, ...
2
votes
2answers
47 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
votes
3answers
27 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 ...
5
votes
3answers
172 views

How to prove inequality $\frac{a}{a+bc}+\frac{b}{b+cd}+\frac{c}{c+da}+\frac{d}{d+ab}\ge 2$

Question: Let $$a,b,c,d>0,a+b+c+d=4$$ show that $$\dfrac{a}{a+bc}+\dfrac{b}{b+cd}+\dfrac{c}{c+da}+\dfrac{d}{d+ab}\ge 2$$ when I solved this problem, I have proved the following three ...
3
votes
2answers
48 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 ...
1
vote
2answers
108 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?
0
votes
2answers
29 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
0answers
192 views
+300

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 ...
8
votes
4answers
1k views

Meaning of $\geqslant$, $\leqslant$, $\eqslantgtr$, $\eqslantless$

What do slanted inequality signs mean? Specifically, these are $\geqslant$, $\leqslant$, $\eqslantgtr$, $\eqslantless$. Is there any place I can look this up? I've searched Wikipedia and the web and ...
1
vote
1answer
33 views

Trigonometric inequality question [on hold]

Let $0 < A < \frac {\pi}{2}$ and $0 < B < \frac {\pi}{2}$. (a) prove that $\sec^2 A + \csc^2 A \cdot \csc^2 B \cdot \sec^2 B \geq 9.$ (b) determine values of $\sec A$ and $\sec B$ when ...
2
votes
2answers
37 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 ...
0
votes
2answers
52 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 ...
0
votes
0answers
19 views

Estimate on the difference of quotients

The following is supposedly true (found it in a paper), however I fail to see why. Let $L(x)$ be a function that goes to $0$ as $x\rightarrow\infty$, $g(n)$ a sequence which goes to $\infty$ as ...
1
vote
1answer
30 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
44 views

Proving Example 1.1.15 of secrets in inequalities

if $a,b,c,d$ are positive real numbers,Prove:$$\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)^2\ge\frac{1}{a^2}+\frac{4}{a^2+b^2}+\frac{9}{a^2+b^2+c^2}+\frac{16}{a^2+b^2+c^2+d^2}$$ ...
4
votes
3answers
53 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 ...
2
votes
2answers
89 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 ...
10
votes
1answer
224 views

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 ...
0
votes
1answer
60 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
vote
3answers
84 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
29 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
1answer
337 views

Proof of Frechet-Hoeffding Copula bounds

How is the lower Frechet-Hoeffding copula bound proved? In the bivariate case, it follows from $C(u_1,u_2)-C(u_1,v_2)-C(v_1,u_2)+C(v_1,v_2)\geq0$ by setting $(v_1,v_2)=(1,1)$. I'm struggling to ...
1
vote
2answers
35 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 ...
1
vote
2answers
50 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 ...
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
53 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 ...
3
votes
1answer
43 views

Cyclic sum inequality involving five numbers with modulus one and zero sum

When working on this MSE question, I was led to conjecture the following : If $z_1,z_2,z_3,z_4,z_5$ are five complex numbers with modulus $1$, such that $z_1+z_2+z_3+z_4+z_5=0$, then $$ ...
-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
36 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 ...
3
votes
4answers
60 views

prove $\sum \limits_{cyc}\frac {a}{(b+c)^2} \geq \frac {9}{4(a+b+c)}$

$a,b,c$ are positive real numbers.prove:$$\sum \limits_{cyc}\frac {a}{(b+c)^2} \geq \frac {9}{4(a+b+c)}$$ Additional info:we cant use induction.we should mostly use Cauchy inequality.other ...
6
votes
2answers
233 views
+50

If $a_1a_2\cdots a_n=1$, then the sum $\sum_k a_k\prod_{j\le k} (1+a_j)^{-1}$ is bounded below by $1-2^{-n}$

I am having trouble with an inequality. Let $a_1,a_2,\ldots, a_n$ be positive real numbers whose product is $1$. Show that the sum $$ ...
1
vote
3answers
2k views

Proving the inequality: $[x-(x^2)/2 < \ln(1+x) < x]$ , $x>0$

Prove the inequality: $[x-(x^2)/2 < \ln(1+x) < x]$ , $x>0$ The right side is easy, I used taylor expansion to show that $e^x > 1+x$ since $e^x = 1 + x + x^2/2 + x^3/3! +\cdots $ The ...
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 $$e^x(x-1)>-2.$$ ...