Questions on proving, manipulating and applying inequalities.

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1
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2answers
29 views

How to prove that $\frac {e^{b^2-1}}{b^2}$ ≥ 1

How to prove that $$\frac {e^{b^2-1}}{b^2} \ge 1?$$ Use logarithm or limit or what? Or do we have to use it as a conclusion to prove it backwards? And how to prove it forwards, that is, without ...
3
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2answers
1k 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 ...
0
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0answers
10 views

Prove $(x+y+z) \cdot \left( \frac1x +\frac1y +\frac1z\right) \geqslant 9 + \frac{4(x-y)^2}{xy+yz+zx}$

$x,y,z >0$, prove $$(x+y+z) \cdot \left( \frac1x +\frac1y +\frac1z\right) \geqslant 9 + \frac{4(x-y)^2}{xy+yz+zx}$$ The term $\frac{4(x-y)^2}{xy+yz+zx}$ made this inequality tougher. It remains me ...
0
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1answer
18 views

Inequality involving the Hessian matrix of a convex function

Let $f \in C^2(\mathbb{R}^d)$ be a convex function with Hessian $H$. Is it true that $$ (x^T H(x) - y^T H(y)) (x-y) \ge 0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1) $$ for all $x,y \in \mathbb{R}^d$? ...
0
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1answer
29 views

For any consecutive natural numbers $a_1,a_2$ are there infinitely many primes $p,q$ such that: $a_1<\dfrac{p}{q}<a_2$?

Progress: Let $a_1,a_2$ consecutive natural numbers; prove or disprove the infinitude of distinct prime pair $p,q$ which satidfies: $a_1<\dfrac{p}{q}<a_2$ The most challenging part of the ...
0
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1answer
30 views

Suppose $0<a<b$. Prove for all $n\geq 2$, $0< \sqrt[n]a< \sqrt[n]b$.

Let a and b be real numbers, and suppose $0<a<b$. Prove for all $n\geq 2$, $0< \sqrt[n]a< \sqrt[n]b$. Proof: Suppose there exists an $n\geq 2$ such that $0 \geq \sqrt[n]a ...
1
vote
1answer
87 views

Prove an inequality using complex analysis

If $f:\mathbb{D}\rightarrow\mathbb{D}$ is holomorphic then prove that $$\frac{|f(0)| - |z|}{1 + |f(0)||z|} \leq|f(z)| \leq\frac{|f(0)| + |z|}{1 - |f(0)||z|} $$ I have been wracking my brain for ...
4
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0answers
25 views

Prove $(x+y)(y^2+z^2)(z^3+x^3) < \frac92$ for $x+y+z=2$

$x,y,z \geqslant 0$ and $x+y+z=2$, Prove $$(x+y)(y^2+z^2)(z^3+x^3) < \frac92$$ While numerical method can solve this problem, I am more interested in classical solutions. I tried this problem for ...
0
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1answer
16 views

How to prove that ${l \choose a_1,…,a_n}\le n^{l-1} $ , when $a_1+…+a_n=l$.

In the proof of (Corollary 8 chap. 3 ) in the book "Sobolev Spaces on Domains" by Burenkov the following inequality is used : given $a_1,...,a_n \in \mathbb{N}$ such that $a_1+...+a_n=l$, then $${l ...
9
votes
2answers
137 views

Prove that $\sin x \cdot \sin (2x) \cdot \sin(3x) < \tfrac{9}{16}$ for all $x$

Prove that $$ \sin (x) \cdot \sin (2x) \cdot \sin(3x) < \dfrac{9}{16} \quad \forall \ x \in \mathbb{R}$$ I thought about using derivatives, but it would be too lengthy. Any help will be ...
29
votes
7answers
779 views

$x,y,z \geqslant 0$, $x+y^2+z^3=1$, prove $x^2y+y^2z+z^2x < \frac12$

$x,y,z \geqslant 0$, $x+y^2+z^3=1$, prove $$x^2y+y^2z+z^2x < \frac12$$ This inequality has been verified to be correct according to Mathematica. $\frac12$ is not the best bound. I try to do AM-GM ...
2
votes
1answer
30 views

Reciprocals of interval union length

Let $I_1,I_2,\ldots,I_n$ be nondegenerate intervals in $[0,1]$. What is the minimum of $\sum_{1\leq i,j\leq n}\frac{1}{|I_i\cup I_j|}$, where the sum is over pairs of intervals that are not disjoint? ...
0
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0answers
9 views

Why is $\left\lvert D^{\alpha}\left(\frac{D\phi}{|D\phi|^2}(\xi)\right)\right\rvert\le C_{\alpha}|\xi|^{-1-|\alpha|}$

Why is $$\left\lvert D^{\alpha}\left(\frac{D\phi}{|D\phi|^2}(\xi)\right)\right\rvert\le C_{\alpha}|\xi|^{-1-|\alpha|}$$ for $\phi\in C^{\infty}(\mathbf R^n)$ sucht that $D\phi(0)=0$, $D\phi\neq0$ ...
0
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0answers
22 views

How to evaluate or approximate this kind of recursion: $a(n+1) = m \cdot \exp(-K \cdot a(n) / m)$?

I came up on a recursive definition of a function, given by $$a(n+1) = m \cdot \exp(-K \cdot a(n) / m),\ n \geq 2$$ with $m$ and $K$ being fixed integers ($m$ large). The first terms of the recursion ...
0
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0answers
14 views

Upper bound for ratio of modified Bessel functions of second kind

I was wondering if someone has an idea if for $0 < x < y$, and $0< \nu \leq \frac{1}{2}$, one can obtain an upper bound for the ratio $$ \frac{K_{\nu}(x)}{K_{\nu}(y)} $$ Thanks.
2
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4answers
48 views

If $x, y, z$ are the side lengths of a triangle, prove that $x^2 + y^2 + z^2 < 2(xy + yz + xz)$

Question: If $x, y, z$ are the side lengths of a triangle, prove that $x^2 + y^2 + z^2 < 2(xy + yz + xz)$ My solution: Consider $$x^2 + y^2 + z^2 < 2(xy + yz + xz)$$ Notice that ...
1
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3answers
70 views

If $a, b, c >0$ prove that $ [(1+a)(1+b)(1+c)]^7 > 7^7a^4b^4c^4 $.

I solved it using AM, GM inequalities and reached to $[(1+a)(1+b)(1+c)]^7 > 2^{21}(abc)^\frac72 $ please help how to get $7^7(abc)^4$ in the inequality.
1
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1answer
22 views

Constructing a determinantal inequality

The following is from page 3410 of the paper Quadratically constrained attitude control via semidefinite programming. Consider a polynomial: $$\mu_1(p_1^Tx)^2+ \cdots + \mu_n(p_n^Tx)^2\leq a$$ ...
0
votes
1answer
26 views

Comparison between my formula and the number 1

It is known that the real numbers $a>0$, $b>0$, and $0<\theta<1$, then is the following inequality correct? $$ \frac{a^\theta\cdot b^{1-\theta}}{\theta\cdot a+(1-\theta)\cdot b}\leq 1 $$ ...
0
votes
1answer
17 views

Prove that $M(|f|,S)-m(|f|,S)\leq M(f,S)-m(f,S)$

Let $M(f,A)=sup\{f(x):x\in A\subseteq[a,b]\}$ and let $m(f,A)=inf\{f(x):x\in A\subseteq[a,b]\}$. Given that $|f(x_{0})|-|f(y_{0})|\leq |f(x_{0}-f(y_{0})|\leq M(f,S)-m(f,S)$ for $x_0,y_0\in S$, prove ...
6
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0answers
193 views
+50

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 ...
2
votes
1answer
71 views

Prove that Triangle ABC is an equilateral triangle iff $\tan{A}+\tan{B}+\tan{C} = 3^\frac32$.

This question is picked from AM GM HM inequalities, so this is to be proved form that concept only, I think it isn't possible because there is no inequality, but if it is please tell me how.
0
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2answers
25 views

Prove that $M_1^2\leq 2M_0M_2$, if $2M_1t≤2M_0+M_2t^2$

Let $0\leq M_1,M_2,M_3\in\mathbb{R}$ and $\forall \ t\in\mathbb{R}:\ 2M_1t≤2M_0+M_2t^2$. Prove that $M_1^2\leq 2M_0M_2$. I tried assigning different values to $t$, but this didn't help.
10
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2answers
2k views

Proof of: $AB=0 \Rightarrow Rank(A)+Rank(B) \leq n$

As the title says, am searching for a proof of If $A,B \in \mathbb{R}^{n\times n}$ and $AB=0$ then $\mathrm{rank}(A)+\mathrm{rank}(B) \leq n$ I am doing this as preparation for an upcoming ...
6
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3answers
1k views

prove that $\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$

If $A$ is a $m \times n$ matrix and $B$ a $n \times k$ matrix, prove that $$\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$$ Also show when equality occurs.
1
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3answers
54 views

Find two fractions such that their sum added to their product equals $1$

This is a very interesting word problem that I came across in an old textbook of mine. So I managed to make a formula redefining the question, but other than that, the textbook gave no hints really ...
0
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1answer
37 views

$\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1 $ then Holder's inequality [duplicate]

If $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1 $ and $ f\in L_p $ $g\in L_q $ and $h\in L_r $ so how can I prove $$ ||fgh ||_1\le||f||_p\ ||g||_q\ ||h||_r $$
0
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3answers
25 views

Inequality for quadratic function and exponent

I might be a but rusty but while doing probability tasks i got stuck on some inequalities from analysis. The task is to prove that there exists $K>0$, so that: $$ \left|\frac{1}{1+e^{3x}} ...
2
votes
1answer
39 views

Deriving a bounding $\delta$ of an interior point

This question is based on the Baby Rudin's 2.16: Regard $Q$, the set of all rational numbers, as a metric space, with $d(p,q)=\lvert p -q \rvert$. Let $E$ be the set of all $p \in Q$ such that $2 ...
1
vote
0answers
16 views

Doob's maximal inequality with stopping time

I have been searching for a version of Doob's maximal inequality with stopping time insides the time index, i.e. given $\Lambda_n$ is a positive sub-martingale and N is a stopping time is there any ...
1
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0answers
26 views

Probability problem related to Markov inequality

Problem Let $p$ be the probability of a person chosen at random to support Bernie Sanders. A sample is taken of $50$ persons chosen at random, each of them is asked if he or she would vote for ...
1
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2answers
45 views

Showing there is a constant for which an inequality holds true

I'm supposed to show that for $x>0$ and $p>0$ there is a constant $C$ such that $e^x\ge Cx^p$. The constant $C$ depends on $p$ but not on $x$. After analysing the behaviour of the graphs of ...
0
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1answer
16 views

Let $x,y \in l_p$, proof that $2^k (||x||^p + ||y||^p)^{2/p} \leq 2 (||x||^2 + ||y||^2)$, where $k = 2 - 2/p$ and $1<p\leq 2$

Let $x,y \in l_p$, proof that $2^k (||x||^p + ||y||^p)^{2/p} \leq 2 (||x||^2 + ||y||^2)$, where $k = 2 - 2/p$ and $1<p\leq 2$ My attempt: It's equivalent to proof the following inequality: $$ ...
0
votes
2answers
44 views

Questions about proof of inequality $(x^p-1)/p <> (x^q-1/q) $

I am working through a proof of the following inequality, where x,p,q are positive, and p and q are integers. $$ \frac{(x^p-1)}{p} \neq \frac{(x^q-1)}{q} $$ Which gives $$ \tag1 ...
0
votes
0answers
34 views

Alignment for equality sign [migrated]

I solved a question and I wrote it in Latex, but the equality sign is not aligned, I searched google, but none of the ways there worked with me. How do I get the equality signs aligned in the below? ...
3
votes
3answers
104 views

A question about $ax = b$

I am studying inequality and come across the following statement. I don't understand it and want to believe the book must have made mistakes. I am going to copy what the book says here exactly. A ...
3
votes
2answers
74 views
+50

Jensen-like averaging inequality on integers

Let $\mathbb{Z}^*=\mathbb{Z}^+\cup\{0\}$. Let $f:\mathbb{Z}^*\rightarrow\mathbb{R}$ be a nondecreasing function such that $f(a+b)\leq f(a)+f(b)$ for all $a,b\in\mathbb{Z}^*$. Is it true that for all ...
2
votes
1answer
35 views

How to prove a set of inequalties in not satisfiable?

For the set of inequalities $$\begin{cases} 10 a - b - c \ge d\\ 5 b - a - c \ge d\\ 2 c - a - b \ge d\\ d \ge a + b + c\end{cases}$$ how can I show these cannot all be satisfied for $a, b, c, d$ ...
0
votes
1answer
14 views

Inequality in recurrence relation

I'm having a mental block understanding what is probably a simple inequality in a guess and check example for a recurrence relation. Would someone please explain to me how they obtain the inequalities ...
1
vote
0answers
58 views

$f \in C^2(\mathbb R)$ , $(f(x))^2 \le 1$ ; $(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 ...
0
votes
0answers
75 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
3answers
88 views

How to prove the inequality $ \frac{1-e^{-x^2}}{x^2}e^{-(x-n)^2}<\frac{2}{n^2}$ for $0<x<n$

Can somebody prove that this inequality is true for $0<x<n$? $$ \frac{1-e^{-x^2}}{x^2}e^{-(x-n)^2}<\frac{2}{n^2}$$ I'm pretty much stuck.
0
votes
0answers
16 views

Inequality for all integers greater than 2 .

Fr all integers $n\geq 2$ , define $f_{n} (x) $ = $(x+1)^{1/n} - x^{1/n}$, where $x>0$ . Then as a function of x a.) $f_{n} (x) $ is increasing for all n b.)$f_{n} (x) $ is a decreasing function ...
0
votes
0answers
14 views

Inequality on exponents of positive numbers

Let x and y be positive numbers . Which of the following always implies $x^y \geq y^x $ a.) $x\leq e\leq y$ b.)$y\leq e \leq x$ c.)$x\leq y \leq e$ or $e\leq y \leq x$ d.)$ y\leq x \leq e$ or ...
1
vote
2answers
122 views

Calculus of variation with inequality constraints

I want to find the function $y$ which maximizes the functional $J[y] = \int_0^1 g(x) y(x) dx$ subject to $0 \leq y(x) \leq 1$ for all $x\in [0,1]$ and $\int_0^1 y(x) dx = k$ where $g$ is a strictly ...
4
votes
2answers
269 views

Is there any solution to find a condition for $f(x)=a+bx^n+cx^2-dx>0$ to always hold true?

Okay, I am interested to know the criteria for a function to always hold $$f(x)=a+bx^n+cx^2-dx>0,$$ if it is given that $a, b, c>0$ and $n\in(-2,2)$ is some real number and $x>0$. My idea ...
3
votes
4answers
108 views

Minimum of $\frac{x}{1+y^2}+\frac{y}{1+x^2}$ on $x,y\ge 0$, $x+y=2$

let $x,y\ge 0$, and such $x+y=2$ find the minimum $$\dfrac{x}{1+y^2}+\dfrac{y}{1+x^2}$$ I think $x=y=1$ is minimum of the value $1$,How can I prove?
0
votes
0answers
29 views

Solving an inequality systematically

A question states: "Find all $n >3$ such that $$ \frac{1}{n^{1.1}}<\frac{1}{n \ln n}" $$ Here's my step: $$ n^{1.1}>n \ln n $$ $$ n^.1>\ln n $$ $$ n >(\ln n)^{10} $$ Setting $(\ln ...
0
votes
0answers
16 views

A inequality concerns with the Legendre polynomial of n-th degree of $\cos\theta$

I am reading a paper, where the author concluded that $${\left| {{P_n}(\cos \theta )} \right|^2} \leqslant \frac{2} {{n\pi \sin \theta }},\,\,\forall \theta \in \left( {0,\pi } \right).$$ Here ...
9
votes
8answers
2k views

Which of the numbers is larger: $7^{94}$ or $9^{91} $?

In this problem, I guess b is larger, but not know how to prove it without going to lengthy calculations. It is highly appreciated if anyone can give me a help. Which number is larger ...