Questions on proving, manipulating and applying inequalities.

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0
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1answer
22 views

$\sup\limits_{t>0}[\frac{g(t)}{\sup\limits_{t<u<\infty}g(u)}]\leq 1$

I can't prove that the following inequality true or not: $$ \sup\limits_{t>0}[\frac{g(t)}{\sup\limits_{t<u<\infty}g(u)}]\leq 1 \tag{*}, $$ where $g$ is a positive function. I think it is true ...
4
votes
1answer
71 views

How do I show that $\frac xy + \frac yz + \frac zx \ge 1 + \frac {z + x}{x + y} + \frac {x + y}{z + x}$?

Show that $$\frac xy + \frac yz + \frac zx \ge 1 + \frac {z + x}{x + y} + \frac {x + y}{z + x}$$ for $x, y, z \gt 0$. I observed that this is a homogeneous inequality so normalization might work. I ...
0
votes
0answers
11 views

Upper bound for ratios of “nearly negative binomial” probabilities

Let $\lambda\in(1/2,1)$, and define an iid sequence of nonnegative random variables $\{X_i\}$ which are "nearly" geometric, in that their distribution behaves similarly to the geometric distributions ...
4
votes
2answers
83 views

Showing a particular type of continuous function is uniformly bounded

Let $I = [0,\infty)$ and $f:I \to I$ be continuous with f(0) = 0. Show that if \begin{equation} f(t) \leq 1 + \frac{1}{10}f(t)^2, \text{ for all } t \in I \end{equation} then $f$ is uniformly bounded ...
0
votes
0answers
24 views

Bounds on $L^2$ and $L^{\infty}$ norms in terms of $H^1$-seminorms for functions attaining a zero in a domain.

We have the following result in one dimension: If $f\in C^1([a,b])$ attains a zero in $[a,b]$, then $$| f |_2 \leq (b-a) | f' |_2$$ and $$ | f |_{\infty} \leq (b-a)^{1/2} | f' |_2$$ Is there a ...
0
votes
1answer
59 views

Suppose $f$ is analytic and $f(a) = f(b) = 0$. Show that $|f(z)| ≤ |{z − a \over 1 − z\bar{a}}| · |{z − b \over 1 − z\bar{b}}|$.

Suppose $f$ is analytic from $D(0, 1)$ to $D(0, 1)$ and $f(a) = f(b) = 0$ for two different numbers $a, b$ in $D(0, 1)$. Show that $\left\vert f(z) \right\vert ≤ \left\vert{z − a \over 1 − z\bar{a} ...
1
vote
1answer
65 views

How do I show that $\frac {\cos^2 A}{\cos^2 B} + \frac {\cos^2 B}{\cos^2 C} + \frac {\cos^2 C}{\cos^2 A} \ge 4(\cos^2 A + \cos^2 B + \cos^2 C)$?

Let $A, B, C$ be the angles of an acute triangle. Show that $$\frac {\cos^2 A}{\cos^2 B} + \frac {\cos^2 B}{\cos^2 C} + \frac {\cos^2 C}{\cos^2 A} \ge 4(\cos^2 A + \cos^2 B + \cos^2 C).$$ How should ...
2
votes
4answers
83 views

How do I show that $\sum_{i = 1}^n \frac 1{\sqrt{a_n}} \lt \frac {\sqrt 3}6$ for $a_n = 4n(4n + 1)(4n + 2)$?

Let $a_n = 4n(4n + 1)(4n + 2)$, show that $$\sum_{i = 1}^n \frac 1{\sqrt{a_i}} \lt \frac {\sqrt 3}6 \quad \forall n \in \mathbb{N}^+.$$ I know I need to find an upper bound for $1/\sqrt{a_n}$ but I ...
2
votes
3answers
83 views

Prove $n^{2n+1}\ge(n+1)^{n+1}(n-1)^{n}$

For natural numbers—that is, integers greater than or equal to 1—prove that: $n^{2n+1}\ge(n+1)^{n+1}(n-1)^{n}$ Equivalently, show that $(1-1/n)^n$ is strictly increasing.
3
votes
3answers
72 views

Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$

I was given a task to prove that inequality is true for x>0: $(e^x-1)\ln(1+x) > x^2$. I've tried to use derivatives to show that the $f(x) = (e^x-1)\ln(1+x)-x^2$ is greater than zero, but has never ...
1
vote
1answer
41 views

How is the following proof really a proof (inequality)?

The user has just subtracted by $\frac{a}{b}$ in the first step and then rearranged the terms to show that it's positive and similar steps have been used to prove the second part of inequality. How ...
0
votes
2answers
79 views

What's wrong with my solution of inequality?

Question: solve the following inequality: $\frac{x}{2} \geq \frac{5}{x + 1} + 4$ My solution: $\frac{x}{2} - \frac{5}{x + 1} + 4 \geq 0$ $\implies \frac{x(x + 1) - 10(x + 1) - 8(x + 1)}{(x + 1) 2} ...
0
votes
1answer
27 views

When should one use a closed interval and when an open one in inequality?

In the following solution: In case I, the person has taken $2x \geq 0$ and then solved the equation. For the other inequality, he has taken $3 - x \gt 0$ and then solved the equation. My question ...
2
votes
1answer
43 views

Find all possible real solutions.

Find all possible real solutions of $a, b, c, d$ and $e$ if: $3a= (b+c+d)^3$ $3b= (c+d+e)^3$ $3c= (d+e+a)^3$ $3d= (e+a+b)^3$ $3e= (a+b+c)^3$ Well I believe the solutions are possible only if ...
2
votes
1answer
16 views

Bounding expectation of a supremum process

This is exercise 3.9(c) on page 15 of Karatzas and Shreve's Brownian Motion and Stochastic Calculus. Let $N_t$ be a Poisson process with intensity $\lambda$. In particular, if $t$ is fixed, $N_t$ is ...
1
vote
1answer
15 views

Solution check: summation inequality proof by induction

I'm not sure if what I've done works or if it's proof enough. (I need to prove that the inequality is true $\forall n \in \mathbb{N}$). $\sum_{i=n}^{2n} \frac{i}{2^i} \leq n$ $P(1)$ works. I assume ...
3
votes
1answer
23 views

Sum greater than 1; minimization for non-strict inequalities

I want to show that if $x_k>0$ for $k=1,2,...,n$ and $\sum_{k=1}^n x_k=1$, then $\sum \frac{x_k^2}{y_k}\ge 1$ for any $y_1, y_2,...,y_n>0$ so that $\sum_{k=1}^n y_k=1$. I tried solving the ...
5
votes
1answer
85 views

Is there an integral that proves that $\sin \tan 1\lt 1$?

I recently noted that this inequality is unbelievably sharp: $$\sin \tan 1\lt 1$$ Is there some sort of integral that can prove that this is true? This question might be of some use: Prove: $\sin ...
0
votes
3answers
23 views

How can we find out the interval in an inequality?

Please go through the following link: Why is equating one of the bracks to zero in this equation correct? Now, the expression given there is $(x+1)(x+3)$, I understand now why we take either of these ...
2
votes
1answer
45 views

Prove inequality: $\frac x{2+xy+yz}+\frac y{2+yz+zx}+\frac z{2+zx+xy}\le \frac{x+y+z}{x+y+z+xyz}$

Numbers $x,y,z$ satisfy $x\in(0,1], y\in(0,1], z\in(0,1]$. Prove inequality: $$\frac x{2+xy+yz}+\frac y{2+yz+zx}+\frac z{2+zx+xy}\le \frac{x+y+z}{x+y+z+xyz}$$ My work so far: $\frac ...
0
votes
1answer
19 views

Functional equation on unit square

Suppose $F$ is a continuous function defined on the unit square $[0,1]\times[0,1]$ satisfying the following properties : i) $ F(a,a)=0,$ for all $a\in[0,1],$ ii) $F(a,b)=-F(b,a)$ for all ...
2
votes
1answer
35 views

Continuous function - $|\int_C f(z)dz| \leq 4$

Question : Let the continuous function $f : \mathbb{C} \to \mathbb{R}$ such that $|f(z)| \leq 1$ and $C$ be the unit circle in the positive direction. Show that $\left|\int_C f(z)dz\,\right| \leq 4$. ...
1
vote
2answers
30 views

Help to solve absolute value inequality

The inequality I have is $\frac {\mid x-1 \mid} {(x+2)} <1 $ what I'm not sure is how I am supposed to proceed. I cannot multiply by (x+2) because it is unknown whether it is positive or negative. ...
1
vote
1answer
40 views

Proving $\frac1{n+1}+\frac1{(n+1)^3} < \frac1n-\frac1{n^3}$

Prove that the sequence of closed intervals $\left[ \dfrac1n-\dfrac1{n^3}~~,~~ \dfrac1n+\dfrac1{n^3}\right]$ for $n>2$ are disjoint. For this to be true I guess we could show that the minimum of ...
1
vote
2answers
41 views

How to solve $\frac{1}{(|x| - 3)}$ $\lt$ $\frac{1}{2}$?

$\frac{1}{(|x| - 3)}$ $\lt$ $\frac{1}{2}$ $x$ can be $\ge$ 0 or $\le$ 0 Case 1 :- $x$ $\ge$ 0 $therefore$, $\frac{1}{(x - 3)}$ $\lt$ $\frac{1}{2}$ $\Rightarrow$ $\frac{1}{(x - 3)}$ - ...
1
vote
0answers
16 views

$\|h\|_{L^{p}} \leq C \|f\|_{L^{p}} \implies \|g\ast h \|_{L^{p}} \leq C_1 \|g\ast f\|_{L^{p}}$?

Suppose that $f, h \in L^{p}(\mathbb R) (1\leq p \leq \infty)$ so that $\|h\|_{L^{p}} \leq C \|f\|_{L^{p}}$ for some constant $C$. Take $g\in \mathcal{S}(\mathbb R^{d})$ (Schwartz Space). We note ...
4
votes
2answers
50 views

Real Analysis triangle inequality problems [duplicate]

$$x,y \epsilon R,\frac{|x+y|}{1+|x+y|}\le\frac{|x|}{1+|x|}+\frac{|y|}{1+|y|}$$ I have to prove this inequality.I used the triangle inequality once and have this step. ...
-1
votes
2answers
42 views

0 as approximate eigenvalue of a matrix [closed]

i got a problem that i cant solve. And i would be grateful for some help. Given the matrix $ X = \begin{pmatrix} 0 & 1 & & \\ -1 & 0 & \ddots & \\ & \ddots & \ddots ...
2
votes
1answer
53 views

Solve inequality $\frac{1}{\sqrt {n+1}} + \frac{1}{n+1}>\frac{1}{\sqrt n} - \frac{1}{n}$ for $3\leq n$

I have to show that although the series $\sum_{n=2}^\infty (\frac{(-1)^n}{\sqrt n} + \frac{1}{n})$ is alternating with the absolute value of the terms approaching zero, it does not contradict the test ...
0
votes
0answers
30 views

If $\frac{1-3\epsilon}{2(1+\epsilon)} \le \cos\theta \le \frac{1+3\epsilon}{2(1-\epsilon)}$, then how does $\theta$ compare to $\frac{\pi}{3}$?

I have $$\frac{1-3\epsilon}{2(1+\epsilon)} \le \cos\theta \le \frac{1+3\epsilon}{2(1-\epsilon)}\qquad\qquad\text{where}\quad 0 < \epsilon < 1$$ What can be said about the value of $\theta$ ...
0
votes
1answer
22 views

Lower bound for third order polynomial over integers

I have the following polynomial: $P(a,b,c,d) = -2 a + 3 a b + 3 a^2 b + 6 a c + 6 b c + 2 d + 3 a d - 3 a^2 d - 3 b d - 6 a b d - 3 d^2 + 3 b d^2 + d^3 \;,$ where $a$, $b$, $c$, $d$ are integers ...
0
votes
4answers
39 views

What is the solution to this inequality?

This is the given inequality I've been trying to solve $$1/6\leq \frac{1}{\mid x \mid} \leq 1/2$$ However the answer I get is $(0,6] \cup [2,6]$ which is not the answer given in my book. Could you ...
1
vote
1answer
48 views

Is this inequality true for all k ? $\sum_{n=k}^{n=+\infty} \frac{1}{n^4} \leq (\sum_{n=k}^{n=+\infty} \frac{1}{n^2})^3$

Can it be generalized for other powers ? Wolfram seems to say it is true for k below 20000. I stumbled upon it randomly when trying to approximate $\sum_{n=1}^{n=+\infty} \frac{1}{n^4}$. My ...
1
vote
0answers
14 views

Azuma's inequality conditional version

Let $(\Omega, \mathcal{F},P)$ be a probability space. Consider a martingale $M_n$ with filtration $\mathcal{F}_n$. Let $B \in \mathcal{F}$. On $B$, $a_n \leq |M_n - M_{n-1}| \leq b_n$ a.s. Can we ...
2
votes
1answer
68 views

Olympiad Inequality AM-GM (easy)

Prove that $(1 + x + y)^2 + (1 + y + z)^2 + (1 + z + x)^2 ≤ 3(x + y + z)^2$, with equality if and only if $x = y = z = 1$ ($xyz \ge 1$) ($x,y,z$ positive reals) This simplifies to $x^2 + y^2 + z^2 ...
2
votes
1answer
58 views

Are primes less than the sum of divisors?

I am trying to prove that Let $p_n$ be the $n$th prime number, $\sigma (n)=\sum_{d|n}d$. Prove that $$\sigma(n) \le p_n$$ It seems obvious at first glance-to me, at least the sum of divisors of ...
6
votes
4answers
110 views

Prove this inequality: $\frac{b+c}{\sqrt{a}}+\frac{c+a}{\sqrt{b}}+\frac{a+b}{\sqrt{c}}\geq \sqrt{a}+\sqrt{b}+\sqrt{c}+3$

Let $a$,$b$,$c$ be positive real numbers such that $abc=1$. Prove that $$\frac{b+c}{\sqrt{a}}+\frac{c+a}{\sqrt{b}}+\frac{a+b}{\sqrt{c}}\geq \sqrt{a}+\sqrt{b}+\sqrt{c}+3$$ I tried various ...
0
votes
3answers
44 views

How to use mean value theorem to prove the inequality $|\sin{x}-\sin{y}|\le|x-y|$ for all $x,y\in\Bbb{R}$?

How to use mean value theorem to prove the inequality $|\sin{x}-\sin{y}|\le|x-y|$ for all $x,y\in\Bbb{R}$? So let us set $f(x)=\sin{x}$ then it's differentiable on $(x,y)$ and continuous on $[x,y]$. ...
5
votes
0answers
136 views

Strict inequality for logarithmic integrals

Let $0<a<b<1$. Does the following inequality hold: $$\max_{f\in L^2[0,a],\,\,\|f\|_2=1}\Bigg|\int_0^a\int_0^af(x)f(y)\ln|x-y|dxdy\Bigg|$$ $$<\max_{g\in ...
1
vote
0answers
41 views

Computing limit using definition of e

I want to show that for fixed $n\in \mathbb{N_0}$, there exists some $C>0$ such that $c>0$ implies $(1-\frac{1}{2^n})^{cn}\le \frac{1}{4c}$. EDIT: Realize now from comments that the limit ...
0
votes
1answer
24 views

Absolute value inequality explanation

I was solving the inequality $-4 \le \left|\frac {x+4} {2-x} \right| \le 4$ and I first wrote the domain, which is $(-\infty,-4] \cup (-4,2) \cup (2, \infty)$ and I got the solution that $x \le \frac ...
1
vote
1answer
19 views

Bounds on derivatives of harmonic functions on unit ball

Let $u$ be a harmonic function on the unit ball in $\mathbb{R}^n$. Show that $$\sup_{B_{1/2}} \lvert \nabla u \rvert \leq C(n) \sup_{\partial B_1} \lvert u \rvert$$ More generally, show that ...
1
vote
0answers
48 views

How to show AM-GM inequality using rearrangement inequality?

Wikipedia article on Rearrangement inequality (link to the current revision) says (without giving any citation for this claim): Many famous inequalities can be proved by the rearrangement ...
5
votes
1answer
52 views

Prove inequality $n\sqrt[n]{n!}-m\sqrt[m]{m!}\le\frac{(n−m)(n+m+1)}2.$

Let $m,n\in\mathbb N$, $n>m$. Prove inequality $$n\sqrt[n]{n!}-m\sqrt[m]{m!}\le\frac{(n−m)(n+m+1)}2.$$ My work so far: $$\sqrt[n]{n!}=\sqrt[n]{1\cdot2\cdot...\cdot ...
0
votes
1answer
32 views

Inequality of arithmetic means of $a_1,a_2,\dots,a_n$ and $a_1,2a_2,\dots,na_n$

Today while reading some text I stumbled upon an inequality which basically boiled down to the validity of the following: If $0\le a_1\le a_2 \le \dots \le a_n$ then ...
0
votes
2answers
82 views

Prove for that $(1 + \frac{x}{p})^p < (1 + \frac{x}{q})^q$ [without Bernoulli inequality or integrals] [duplicate]

Prove for $x > 0$ and $0 < p < q$ that $$(1 + \frac{x}{p})^p < (1 + \frac{x}{q})^q$$ I think that binomial theorem might be of use in this exercise, but I'm not sure. I haven't been able ...
0
votes
2answers
49 views

How to show that $3(a^4+b^4)+2a^4b^4\leq8$ if $a^3+b^3=2$

It is given that a and b belong to non negative real numbers and $a^3+b^3=2$. Then prove that $3(a^4+b^4)+2a^4b^4\leq8$
1
vote
1answer
44 views

Some inequalities between 1- norm, 2- norm and infinity-norm: $\|x\|_2\leq\sqrt{\|x\|_1\| x\|_\infty}\leq\frac{1+\sqrt{n}}{2}\|x\|_2$

Let $x\in\mathbb{C}^n$. Do the following inequalities hold? $$\lVert x\lVert_2\leq\sqrt{\lVert x\lVert_1\lVert x\lVert_\infty}\leq\frac{1+\sqrt{n}}{2}\lVert x\lVert_2.$$ I think the first inequality ...
1
vote
1answer
23 views

Calculating the Legendre transform of a function explicitly

This is a portion of one of the questions in Evan's PDE book. Let $H: \mathbb{R}^n \to \mathbb{R}$ be defined by $H(p) = \frac{1}{r}|p|^r,$ for $1 < r < \infty$. I want to show that $$L(q) = ...
0
votes
2answers
59 views

proof of an inequality $ \left( 1+ \frac{x}{p} \right) ^{p}$

Prove that for $x>0$ and $0<p<q$: $$\left( 1+ \frac{x}{p} \right) ^{p} < \left( 1+ \frac{x}{q} \right) ^{q} $$