Questions on proving, manipulating and applying inequalities.

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0
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1answer
43 views

Prove that $\forall \varepsilon\gt 0\ \ \exists N\in 2\mathbb N$ so that the inequality $2+\frac{1}{N}\lt \frac{9}{5} + \varepsilon$ stands

Prove that $\forall \varepsilon\gt 0\ \ \exists N\in 2\mathbb N$ so that the inequality $2+\frac{1}{N}\lt \frac{9}{5} + \varepsilon$ stands. I tried to go from this expression to another one that ...
6
votes
1answer
61 views

The number of positive integer solutions to the equation $x_1+2x_2+…+nx_n=n^2.$

Let $n \ge 2, n \in \mathbb N$. $A_n$ denotes the number of positive integer solutions to the equation $$x_1+2x_2+...+nx_n=n^2.$$ Prove inequality $$\frac{n^n(n-1)^{n-1}}{2^{n-1}\left(n!\right)^...
1
vote
1answer
23 views

Proving an inequality about a set of combinations.

Suppose $A$ is a set of $r$ combinations of an $n$ set, with $\alpha \cap \beta \neq \phi$, whenever $\alpha, \beta \in A$. Show that $$|A| \leq \binom{n-1}{r-1}$$ if $r \leq \frac n2$. What does ...
0
votes
2answers
68 views

Prove $\frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$.

So I have to prove $$ \frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}.$$ I rearranged it $$ a^2bc + ab^2c + abc^2 \leq b^2c^2 + a^2c^2 + a^2b^2 .$$ My idea from there is ...
0
votes
2answers
26 views

Inequality involving a convex function

I am stuck, showing the following inequality in an easy way (using only inequalities or something): Let $x\in [-a,a]$ for some $a>0$ and $p\in (1,2)$. I want to show that there then exists a ...
1
vote
1answer
15 views

Prove that $a^2 pq + b^2 qr + c^2 rp \leq $ given a,b and c are sides of triangle and p+q+r=0

The question is asking to prove that $a^2 pq + b^2 qr + c^2 rp \leq 0 $ given that $a,b$ and $c$ are the sides of a triangle and that $p+q+r=0$. I have tried AM GM as well as countless pages of ...
4
votes
5answers
526 views

Intuition behind Chebyshev's inequality

Is there any intuition behind Chebyshev's inequality or is that only pure mathematics? What strikes me is that any random variable (whatever distribution it has) applies to that. $$ \Pr(|X-\mu|\...
0
votes
1answer
18 views

Integral Inequality with Monotonic Function

Problem: For continuous, either both increasing or both decreasing functions $f, g$ on $[a, b]$, suppose that $p(x)$ is continuous and positive. Prove that $$\int_a^bp(x)f(x)dx \int_a^bp(x)g(x)...
0
votes
1answer
90 views

Inequality $ab\le \frac{a^p}{p}+\frac{b^q}{q}$ [duplicate]

If $\frac {1}{p}+\frac {1}{q}=1$ and $a,b \ge 0$ , then prove $ab\le \frac{a^p}{p}+\frac{b^q}{q}$ . I can't find a simple and short way to prove this. Any hint would work. Thanks in advance!
1
vote
2answers
86 views

Show the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle

So I need a little help with the following: Considering separately the cases of real and complex roots show that the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle (i.e....
0
votes
1answer
13 views

Why is $|\psi_n-f|^p \leq2^p |f|^p$ when $|\psi_n|\leq |f|$?

Why is $|\psi_n-f|^p\leq 2^p |f|^p$ when $|\psi_n|\leq|f|$? Is it true that $|a+b|^p\leq 2^p (|a|^p+|b|^p)$?
2
votes
2answers
42 views

Inequality involving ArcTan

How to prove that for $x\in[0, +\infty]$ the following inequality is true: $$\arctan x\geq\frac{3 x}{1+2\sqrt{1+x^2}}?$$ I don't have idea from where to start, so any hint is welcome. Thanks in ...
3
votes
1answer
46 views

Does this inequality involving inverse tangent (arctan) hold?

I am wondering if the following statement is true for $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ and $x,y\in\mathbb{R}$: $$\tan^{-1}\left(\frac{\sin(\theta)+x}{\cos(\theta)+y}\right)\leq\...
7
votes
3answers
94 views

Prove inequality $\sqrt{\frac{1}n}-\sqrt{\frac{2}n}+\sqrt{\frac{3}n}-\cdots+\sqrt{\frac{4n-3}n}-\sqrt{\frac{4n-2}n}+\sqrt{\frac{4n-1}n}>1$

For any $n\ge2, n \in \mathbb N$ prove that $$\sqrt{\frac{1}n}-\sqrt{\frac{2}n}+\sqrt{\frac{3}n}-\cdots+\sqrt{\frac{4n-3}n}-\sqrt{\frac{4n-2}n}+\sqrt{\frac{4n-1}n}>1$$ My work so far: 1) $$\...
3
votes
3answers
43 views

Prove that $\frac{1}{4\cdot 1976^3}-\frac{1}{16\cdot 1976^7}>10^{-19.76}$

Prove that $\frac{1}{4\cdot 1976^3}-\frac{1}{16\cdot 1976^7}>10^{-19.76}$ without using a calculator. I rearraged to get $4 \cdot 1976^4-1 > 10^{-19.76} \cdot 16 \cdot 1976^7$ and so we have ...
1
vote
3answers
46 views

Prove that $\sqrt{n+1}>\sqrt{n}+\frac{1}{2\sqrt{n}}-\frac{1}{8n\sqrt{n}}$

Prove that $\sqrt{n+1}>\sqrt{n}+\frac{1}{2\sqrt{n}}-\frac{1}{8n\sqrt{n}}$ if $n>0$. I didn't see an easy way of proving this without doing a lot of algebra and rearranging. Is there an easier ...
0
votes
1answer
13 views

Trace class norm and rank inequality

I am quite new to operators in Hilbert spaces and I have been trying to show that for any linear and bounded operator $T : \mathcal{H} \rightarrow \mathcal{H}$ \begin{equation} \vert \vert T \vert \...
0
votes
0answers
15 views

Uniform Boundary for S.D.E with Lipschitz Coefficients

Let $X_t^x$ be a solution to the SDE: $dX_t=b(X_t)+\sigma(X_t)dW_t$ with $X_0=x$. Assume that $b$ and $\sigma$ are Lipschitz Continuous. I want to proof that there exists $0<L$ such that $E\left|...
-3
votes
1answer
32 views

Neighbourhood set in Graph theory [on hold]

Let $G$ be any connected graph with $\Delta(G)$ be maximum degree. If $D \subseteq V(G)$ then how can we say that $\left | \bigcup \limits_{v \in D} N(v) \right | \leq |D| \Delta(G)$.
62
votes
8answers
5k views

What is Cauchy Schwarz in 8th grade terms?

I'm an 8th grader. After browsing aops.com, a math contest website, I've seen a lot of problems solved by Cauchy Schwarz. I'm only in geometry (have not started learning trigonometry yet). So can ...
1
vote
3answers
43 views

Solutions to the inequality $0>-x^2 +2x+3.$

I am trying to solve an inequality of $$0>-x^2 +2x+3.$$ I am aware of two different ways of factorizing this. $$(-x+3)(x+1)\quad\text{ and }\quad(x-3)(-x-1).$$ When I use $(-x+3)(x+1)$, I get the ...
1
vote
4answers
36 views

$\cos2\theta +\cos\theta +k = 0 $ - set of all values of $k$ for which there is a solution

The set of all values of $k$ (real), such that the equation $\cos2\theta +\cos\theta +k = 0 $ admits a solution for $\theta$ is? MY ATTEMPT: I substituted $\cos2\theta$ with $2\cos^2\theta - 1 $. On ...
0
votes
1answer
30 views

Would the following series of implications be logically correct?

Let $a$ and $b$ be positive integers, and let $f$ be a generic function satisfying $f(1) = 1$, and taking on only positive integer values. Suppose that I have the following propositions: $$\bf{A} : ...
5
votes
5answers
111 views

Bounding a series: $\frac{\pi}{2} < \sum_{n=0}^\infty \frac{1}{n^2 + 1} < \frac{3\pi}{2} $

I have the following statement - $$\frac{\pi}{2} < \sum_{n=0}^\infty \dfrac{1}{n^2 + 1} < \frac{3\pi}{2} $$ So I tried to prove this statement using the integral test and successfully proved ...
1
vote
1answer
39 views

It's true that $ |\log^2(z)| \leqslant |\log(R)|^2 + |i \arg(z)|^2 $ where $z \in \mathbb{C}$

In some residue integral, when one have to prove that an integral vanish at infinity, I've found in some textbooks the inequality: $$ |\log^2(z)| \leqslant |\log(R)|^2 + |i\ \arg(z)|^2 $$ Where $z= ...
2
votes
0answers
21 views

Finding a maximum with some constraints

I would like to maximize the term $ l_1b_1+l_2b_2+l_3b_3-2 $ such that the following conditions hold: $ 1>l_1>l_2>l_3>0 $, $ l_1,l_2,l_3 \in \mathbb{Q} $, $ b_1,b_2,b_3 \in \mathbb{N} $...
0
votes
0answers
15 views

Convex hull possesses only integer extreme points

I have the following question. Consider given natural numbers $ 1 \le l_m <\ldots < l_1 < L $. Is it possible to prove that the convex hull of $ \left\lbrace a \in \mathbb{Z}^m_{\ge 0} \, \...
2
votes
0answers
94 views
+50

$f \in C^2(\mathbb R)$ , $(f(x))^2 \le 1$ ; $(f'(x))^2+(f''(x))^2 \le 1 $ ; then is $(f(x))^2+(f'(x))^2 \le 1 $?

Let $f \in C^2(\mathbb R)$ be such that $$(f(x))^2 \le 1 ; (f'(x))^2+(f''(x))^2 \le 1 , \forall x \in \mathbb R$$ Then is it true that $(f(x))^2+(f'(x))^2 \le 1 , \forall x \in \mathbb R$ ? I ...
2
votes
3answers
134 views

Show that $|A+A|\geq (2n-1)$

Consider a set $A$ consisting of $n$ natural numbers $\{a_i\}_{i=1}^n$ such that $a_1<a_2 < \cdots <a_{n-1} < a_n$. Define the set $A+A$ such that it contains $a_i + a_j \ ; \ i \leq j$ as ...
2
votes
2answers
53 views

How to find the restrictions of side length on an obtuse triangle

Question: In Triangle ABC, the angle ∠ABC is an obtuse angle. The Side AB is 1cm, and the side BC is 3cm. Side AC is (3x+10)/(x+3) cm Find the restriction(s) on x. I have tried a few different ...
4
votes
1answer
103 views

Approximately not equal

What terms do you consider appropriate for the relations denoted by symbols like these: $$\Large 1.≈\qquad 2.≉\qquad 3.⪅\qquad 4.⪉$$ The first one should be easy: “almost equal to” and “...
0
votes
1answer
52 views

Special Case of A.M-G.M Inequality. [on hold]

How can we prove the special case of A.M-G.M Inequality, that is: The Geometric Mean of $n$ positive real numbers is equal to 1.Prove that their Arithmetic Mean is greater than or equal to 1. I ...
0
votes
1answer
23 views

Automatic tools for solving a set of inequalities

I am trying to solve problems such as the following: Find real numbers $a_1,a_2,a_3, b_1,b_2,b_3$ such that all following expressions are true: $a_1+a_2+a_3=0$ $b_1+b_2+b_3=0$ $a_1>...
0
votes
1answer
35 views

Prove that $-4\leq5\cos\theta+3\cos(\theta+\frac{\pi}{3})+3\leq10$

Prove that $$\color\red{-4}\leq5\cos\theta+3\cos(\theta+\frac{\pi}{3})+3\leq\color\red{10}$$ My attempt:- I simplified the equation to $$\begin{align} &\;\;\phantom{=} 5\cos\theta+3\cos(\...
7
votes
7answers
203 views

How to prove $\frac xy + \frac yx \ge 2$

I am practicing some homework and I'm stumped. The question asks you to prove that $x \in Z^+, y \in Z^+$ $\frac xy + \frac yx \ge 2$ So I started by proving that this is true when x and y have ...
0
votes
1answer
31 views

Variant of Holder's inequality: $\|x\|_p \le n^{\frac1p- \frac1r} \|x\|_r$

So far I believed that only the reverse Holder inequality holds for $0<p<r<1,$ but then a student pointed out to me that $$\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$$ A few numerical ...
147
votes
2answers
7k views

How does one prove the matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrices $A,B,C\in M_{n}(C)$ be Hermitian and Positive definite matrices, such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ ...
1
vote
2answers
102 views

prove inequation

a,b,c,d $\in \mathbb{R}$ $a,b,c,d \gt 0$ and $ c^2 +d^2=(a^2 +b^2)^3$ prove that $$ \frac{a^3}{c} + \frac{b^3}{d} \ge 1$$ If I rewrite the inequation like $ \frac{a^3}{c} + \frac{b^3}{d} \ge \frac{...
0
votes
1answer
46 views

Cauchy-Schwarz inequality application

I am having trouble verifying the first step where the author makes use of Cauchy-Schwarz inequality. https://gbas2010.wordpress.com/2011/10/16/inequality-53-vo-quoc-ba-can/ I am unsure how he ...
8
votes
2answers
665 views

Prove $\sum_{cyc}\left(\frac{a^4}{a^3+b^3}\right)^{\frac34} \geqslant \frac{a^{\frac34}+b^{\frac34}+c^{\frac34}}{2^{\frac34}}$

When $a,b,c > 0$, prove $$\left(\frac{a^4}{a^3+b^3}\right)^{\frac34}+\left(\frac{b^4}{b^3+c^3}\right)^{\frac34}+\left(\frac{c^4}{c^3+a^3}\right)^{\frac34} \geqslant \frac{a^{\frac34}+b^{\frac34}+c^{...
1
vote
1answer
55 views

How to prove this inequality $x\sin^2{A}+y\sin^2{B}\ge xy\sin^2{C}$

In $\Delta ABC$,if $x,y>0$ and $x+y=1$.show that $$x\sin^2{A}+y\sin^2{B}\ge xy\sin^2{C}$$ I have looked at the simpler methods,? Here is one solution $$\dfrac{\sin^2{A}}{y}+\dfrac{\sin^2{B}}{...
0
votes
1answer
45 views

Matrix Trace Inequality [on hold]

If $\operatorname{Tr}(A) < \operatorname{Tr}(B)$, is it fair to say that $\operatorname{Tr}(AC) < \operatorname{Tr}(BC)$? All of $A$, $B$ and $C$ are positive definite matrices.
2
votes
1answer
136 views

How to prove that $\frac{x^2+y^2}{4}\leq e^{x+y-2}$

I need to prove that given any $x,y\in \mathbb{R}$, such that $x,y\geq 0$, then is true that $$\frac{x^2+y^2}{4}\leq e^{x+y-2}$$ My try was use logarithms and state that without losing generality, ...
2
votes
1answer
46 views

Orthogonal matrix $Q$ such that $\forall x\leq 0$, $Qx\geq 0$

What are the orthogonal matrices $Q$ such that for all vectors $x\leq 0$, $Qx\geq 0$? The inequality is to be understood component-wise. In dimension 1, the only possibility is $Q=[-1]$, which is a ...
3
votes
1answer
59 views

Functional inequality $\sum_{1\le i<j\le n}f(x_i+x_j)\ge \frac{n(n-1)}{2}f(a_1x_1+a_2x_2+…+a_nx_n)$

Let $n\in\mathbb N, n\ge 2$. Does there exist a set of non-zero real numbers $a_1, a_2,..,a_n$ with this condition: If function $f: \mathbb R \rightarrow \mathbb R$ satisfies the inequality $$...
3
votes
7answers
3k views

Prove $n^2 > (n+1)$ for all integers $n \geq 2$

I understand that I need to use induction for this, that's not a problem. I get stuck after I try to invoke the inductive hypothesis. $P_n: n^2 > n+1$... and we want to prove $P_{n+1}: (n+1)^2 >...
1
vote
2answers
70 views

How can I solve this inequality? [on hold]

Have a nice day, how can I solve this inequality? $$a<b<-1$$ $$ |ax - b| \le |bx-a|$$ what is the solution set for this inequality
0
votes
1answer
129 views

Prove or disprove $2abc(a+b+c)\ge 3(a^2b^2c^2+1)$

Let $a,b,c>0,ab+bc+ca=3$, prove or disprove $$2abc(a+b+c)\ge 3(a^2b^2c^2+1)$$ Now I can't find any counterexample
-2
votes
0answers
31 views

Probability: How do I prove this inequality? [duplicate]

While studying probabilities and I have encountered this inequality. I'm trying to prove that for any random variable $X$ and any $\epsilon \gt 0$ this inequality is correct. $$P(|X - EX| \ge \...