Questions on proving and manipulating inequalities.

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0
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3answers
37 views

a simple inequality

Is it true that for any real numbers (a,b): $(a - b)^{2} \leq 3a^{2} + 3b^{2}$ Also, if this is true, is there a way to sharpen this bound say $(a - b)^{2} \leq K(a^{2} + b^{2})$, for some $K < ...
4
votes
3answers
84 views

Prove the inequality $({1+\frac{a}b})^n$ + $(1+\frac{b}a)^n$ $\geq$ $2^{n+1}$

Let $a$ and $b$ be positive real numbers and let $n$ be a natural number prove that $$\left({1+\frac ab}\right)^n+\left(1+\frac ba\right)^n\ge2^{n+1}.$$
1
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2answers
49 views

Proving that the line integral $\int_{\gamma_{2}} e^{ix^2}\:\mathrm{d}x$ tends to zero

Let $f(z) = e^{iz^2}$ and $\gamma_2 = \{ z : z = Re^{i\theta}, 0 \leq \theta \leq \frac{\pi}{4} \} $. All the sources I have found online, says that the line integral $$ \left| \int_{\gamma_2} ...
2
votes
0answers
32 views

Integral inequality related to derivation

While trying to understand a proof, i have stumbled upon the following statement: Let $f \in L^p(a,b)$ be a $p$-integrable function. Then the inequality $$\liminf_{s \rightarrow t} \frac{1}{t-s} ...
0
votes
2answers
47 views

Inequality with moments

Let $m$ a probability measure, $f$ a positive measurable function (one can assume it is bounded, the existence of the moments is not a problem here). Is $m(f^3) \le m(f^2) m(f)$?
8
votes
4answers
2k views

Proof of Bernoulli's inequality

The question reads $$U_n = (1+x)^n - 1 - nx$$ Show that $U_2 \geq 0$ Hence or otherwise show that $(1+x)^n \geq 1 + nx$ for all $x \gt -1$. Obviously the $U_2 \geq 0$ is very easy, I can do ...
1
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2answers
72 views

Positivness of the sum of $\frac{\sin(2k-1)x}{2k-1}$.

For $n\in \mathbb{N}$, $x\in (0,\pi)$. Prove that : $$f_n(x)=\sum_{k=1}^n \frac{\sin [(2k-1)x]}{2k-1} \geq 0.$$ I've tried to do it by differentiation : I Calculate $f_n'(x)$ (sum of ...
1
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1answer
27 views

Prove trace inequality $\mathrm{tr}\{ABCBAD-ABCD-ADCB+CD\} \geq 0$

Let $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, $\mathbf{D}$ be four (generally non-commuting) positive semidefinite matrices of same size. I want to show that (or find a counterexample to) $$ ...
1
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1answer
20 views

For $1-r<|\theta|<1/2$, $|\frac{2r\sin{2\pi\theta}}{1-2r\cos{2\pi\theta}+r^2}-\frac{\cos\pi\theta}{\sin\pi\theta}|<C\frac{(1-r)^2}{|\theta|^3}$

For $1-r<|\theta|<1/2$ show that $$|\frac{2r\sin{2\pi\theta}}{1-2r\cos{2\pi\theta}+r^2}-\frac{\cos\pi\theta}{\sin\pi\theta}|<C\frac{(1-r)^2}{|\theta|^3}$$ This inequality shows that ...
0
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1answer
27 views

How to use induction on this type of inequality?

Given $a_1,a_2,\ldots,a_n>0$ and $a_1+a_2+\ldots+a_n<\frac{1}{2}$, prove that $(1+a_1)(1+a_2)\ldots(1+a_n)<2$. Some of you may have already seen this inequality. I was the one who asked ...
3
votes
0answers
57 views

Rising Sun Inequality (Dunford-Schwartz maximal inequality)

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
0
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4answers
81 views

Proof that $(1 + x)^n > 1 + nx$ for $x>-1$, $n$ a positive integer [duplicate]

For any positive integer $n$ and real number $x > -1$, show that $(1 + x)^n > 1 + nx$. This is Bernoulli’s inequality but I can't figure out how to start with this. Can someone help? Thanks
0
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3answers
65 views

Is there any book about inequality? [closed]

I heard there is a book name 'inequality'. But I couldn't find the book. Is there any site or book about inequalities? What i want is collection of inequalities.
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0answers
12 views

Inequality about ceiling function. How to prove it?

Proposition: Set $n,m\in{\mathbb{Z}}$ with $n\geq{2}$, then $$\lceil{\frac{n}{3}}\rceil\lceil{\frac{n}{3}}\rceil\leq\lceil{\frac{nm}{5}}\rceil$$ I've verified it for small cases (computationally), ...
1
vote
3answers
79 views

Have I justified that $\forall x \in \mathbb{R}$, $x > 1 \rightarrow x^2 > x$

Have I justified that $\forall x \in \mathbb{R}$, $x > 1 \rightarrow x^2 > x$ Here is what I would do if this were asked on a test and I was told to "justify" the answer. Let $x \in ...
0
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1answer
41 views

Induction proof that $4^n > 3^n+2^n$ for $n\ge2$

This is a problem with induction and proofs but I'm not sure how to start with proving this one. $$\text{Show that for any $n \geq 2$, $4^n > 3^n+2^n$}$$
0
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3answers
63 views

Showing that $1+a>0 \implies (1+a)^n \ge 1 + na$ [duplicate]

Establish the Bernoulli Inequality if $1+a>0$, then $(1+a)^n\ge 1+na$ for $n\ge 1.$ So far I have $$(1+a)^k+1 \ge 1+(k+1)a,$$ but I don't know what to do from here to make the left and right ...
1
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6answers
48 views

$c \le n < m < c+1 \implies m-n<1 $

This should be simple, but I got stuck trying to manipulate the inequalities. Show that: $c \le n < m < c+1 \implies m-n<1 $, all are real numbers. I have that $m< c + 1 \implies ...
5
votes
2answers
56 views

How to show $a+b+ad\geq c+d+bc$ given $a\geq c$ and $a+b\geq c+d$?

Let $0\leq a,b,c,d\leq 1$ and $a\geq c$ and $a+b\geq c+d$. Show that $a+b+ad\geq c+d+bc.$ Of course we have $a+b\geq c+d$, but how to relate $ad$ and $bc$?
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1answer
19 views

Proving elementary inequalities with equations

Assume $b > 0,\ d > 0$. Assume: $$ \frac{a}{b} < \frac{c}{d} $$. Prove that: $$ \frac{a}{b} < \frac{a + c}{b + d} < \frac{c}{d} $$. I would like to find an intuitive way to solve ...
14
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5answers
2k views

Inequality: $(x + y + z)^3 \geq 27 xyz$

Edit: $a,b,c$ and $x,y,z$ are positive, real numbers. Since $(a-b)^2 \geq 0~$, $a^2 + b^2 - 2ab\geq0~$ and $a^2 + b^2 \geq 2ab~$. Similarly, $a^2 + c^2 \geq 2ac~$ and $b^2 + c^2 \geq 2bc~$. ...
1
vote
1answer
55 views

Relation between softmax and max

For two vectors $X$ and $Y$ in $\mathbf{R}^n$, does the inequality below hold? $\left| \text{softmax} X - \text{softmax} Y \right| \leq \text{max} | X - Y |$ Softmax is the same as log-sum-exp: ...
6
votes
2answers
84 views

Prove that positive $x,y$ satisfy $\left(\frac{1}{1+x}\right)^2+\left(\frac{1}{1+y}\right)^2\ge\frac{1}{1+xy}$.

Prove that positive $x,y$ satisfy $$\left(\frac{1}{1+x}\right)^2+\left(\frac{1}{1+y}\right)^2\ge\frac{1}{1+xy}$$ My teacher claims this lemma is often useful. I'm wondering, though: how to prove ...
2
votes
1answer
186 views

if $a_n=\frac{a_{[\frac{n}{2}]}}{2}+\frac{a_{[\frac{n}{3}]}}{3}+\ldots+\frac{a_{[\frac{n}{n}]}}{n}$,then $a_{2n}<2a_{n}$

Question: Consider the following sequence : $$a_1=1 ; a_n=\frac{a_{[\frac{n}{2}]}}{2}+\frac{a_{[\frac{n}{3}]}}{3}+\ldots+\frac{a_{[\frac{n}{n}]}}{n}$$. Prove that: $$a_{2n}< 2a_{n } (\forall ...
1
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2answers
39 views

Number of solutions for inqeuality

Is there a way we can determine number of solutions for equation $$x*y < d$$ where d is constant and x & y are positive integers greater than 1. I am not interested in actual values, but ...
0
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4answers
118 views

Does $xy\geq x+y$?

I just see the GM-AM inequality. But I would like to compare $xy$ with $x+y$ for any $(x, y)\in\mathbb{R}^2$. It looks like $xy>x+y$ since the first one is multiplication and the second one is ...
2
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1answer
19 views

Inequality with two moduli

I have a question similar to this, find all $x \in \mathbb{R}$ satisfying $\displaystyle 3 < \left| x+1 \right| + \left| x - \frac{1}{2} \right| < 7$ which is rather trivial by distinguishing ...
1
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1answer
25 views

Is there another way to solve the value field of a parameter of an line.

Assume $P$ is a point in line $x+y=m$, where $m \in \Bbb{R}$. There are two points $A,B$ in circle $$x^2+y^2 = 10$$ such that $PA$ and $PB$ are tangent lines of the above circle. If line: $x+y=m$ has ...
2
votes
3answers
37 views

Find such anti-symmetric matrix $W$ that $A^T WP \geq 0$

$P$ and $A$ are both n-dimensional vectors with non-negative components. $W$ is an $n\times n$ matrix with $W_{ij}=w_i-w_j$, where all $w_k\geq 0$. So $W$ is an anti-symmetric matrix with some ...
2
votes
2answers
42 views

Prove that positive $x,y,z$ satisfy $\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}\ge \sqrt{2x}+\sqrt{2y}+\sqrt{2z}$

Prove that positive $x,y,z$ satisfy $$\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}\ge \sqrt{2x}+\sqrt{2y}+\sqrt{2z}$$ Actually, this is a part of my solution to another problem, which is as given below: ...
1
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1answer
32 views

Show $|xy-x_0y_0|<\epsilon$: necessary assumptions on $x,y$

From Spivak's Calculus, problem 1.21: Prove that if $$|x-x_0| < \min \left( \frac{\epsilon}{2(|y_0|+1)},1 \right)\qquad |y-y_0|< \frac{\epsilon}{2(|x_0|+1)}$$ then $|xy-x_0y_0|<\epsilon$. ...
2
votes
3answers
54 views

How to prove this inequation?

$$ 1+\frac{2}{3n-2}\leqslant \sqrt[n]{3}\leqslant 1+\frac{2}{n}, n\in \mathbb{Z}^{+} $$ How to prove this inequation?
2
votes
2answers
48 views

If positive $a,b,c,d$ satisfy $(a^3+b^3)^4=c^3+d^3$, prove that $a^4c+b^4d\ge cd$.

If positive $a,b,c,d$ satisfy $(a^3+b^3)^4=c^3+d^3$, prove that $$a^4c+b^4d\ge cd$$ It kind of seems useful to begin with a division of both sides by $cd$: $$\frac{a^4}{d}+\frac{b^4}{c}\ge1$$ It ...
1
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1answer
39 views

Solve the inequality…

Can you please show me how can I solve this inequality. I would like to see how it can be done without the graph of the functions. Thank you! $$2\sqrt{(x-1)(x+2)}\ge|x+1|-2$$
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1answer
48 views

Solve the following inequality…

Can you please verify if I've done this exercise correctly, and if you have a better solution, please, show it to me. Thank you! (The exercise is in the left top corner.)
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0answers
12 views

Schur-concave functions, derivative sign help

To establish some inequality I must prove: $$\dfrac{\partial}{\partial ...
0
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0answers
16 views

Inequality with nesting floors and ceilings

EDIT: I changed the inequality to a simpler more indicative format. I need to solve the following inequality for $x$ (find the minimum value of $x$) but I have trouble removing the ceiling and floors ...
1
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1answer
29 views

How find this value of $x$ such $\log_{\frac{1}{12}}{(x^2+2x-3)}>x^2+2x-16$

if $$\log_{\frac{1}{12}}{(x^2+2x-3)}>x^2+2x-16$$ Find the value of $x$ My idea: since $$x^2+2x-3>0\Longrightarrow x>1 ,or, x<-3$$ ...
3
votes
3answers
44 views

Inequality Exercise in Apostol's Calculus I

Let p and n denote positive integers. Show that: $$n^{p} \lt \frac{(n+1)^{p+1} - n^{p+1}}{p+1} < (n+1)^{p}$$ Attempt at Solution Using the identity $b^{p+1}-a^{p+1} = ...
4
votes
2answers
71 views

Show that $f(x_0)-f(x)<\vert x_0-x\vert$ for all $x \ne x_0$

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ continuous and bounded below. Show that there exist $x_0 \in \mathbb{R}$ such that $\forall x\ne x_0$, $$f(x_0)-f(x)<\vert x_0-x\vert$$ Since $f$ ...
2
votes
0answers
50 views

Inequality, symmetric function

Let $$F(x,y)=\dfrac{f\left(\frac{x}{x+y}\right)+f\left(\frac{y}{x+y}\right)}{x+y},$$ where $f>0$ is a concave function. Using brute force computation (computer based proof) with ...
0
votes
4answers
62 views

Elementary proofs of inequalities

I was just introduced into elementary proofs of inequalities, my text's explanation however feels incomplete. I did further research on the subject, my question is thus: Prove: If $0 < a < b$, ...
7
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1answer
218 views

A $2\times2$ Matrix inequality

$M,N$ are $2\times2$ real matrices, and $MN=NM$. Then, for any three real numbers $x,y,z$, we have $$4xz\det(xM^2+yMN+zN^2)\geq(4xz-y^2)\big(x\det(M)-z\det(N)\big)^2 $$ some thought: 1). ...
1
vote
1answer
36 views

AM-GM inequality

On the wikipedia page on "Nesbit's inequality", the fifth proof ends as follows: $$ \frac{x+z}{y}+\frac{y+z}{x}+\frac{x+y}{z}\geq 6$$ which is true, by AM-GM inequality. I am wondering if the ...
2
votes
0answers
22 views

Muirhead's Inequality (software?)

I just started learning about inequalities: Schur's, Karamata's, Muirhead's, etc... They are beautiful but it seems that in the case of more than two variables, some of the computations become a ...
0
votes
1answer
22 views

An integral inequality question.

If we have two functions $f,g:[a,b]\to\mathbb{R}$ and we know they are bounded, so: $\sup_{x\in[a,b]}|f(x)|=K$, and $\sup_{x\in[a,b]}|g(x)|=M$. Where $K,M$, are positive finite constants, which of ...
1
vote
1answer
28 views

How to prove lower and upper bound for exponential sum?

A paper I'm reading implicitly uses the fact $$\sum\limits_{t=1}^n e^{-ta^2} \in \theta(\frac{1}{a^2})$$ (It uses the both $\leq$ and $\geq$ sides in the proofs). I'm able to prove that ...
3
votes
3answers
121 views

Prove that, for any positive integer n: $(a + b)^{n} \leq 2^{n-1}(a^{n}+b^{n})$

Prove that, for any positive integer n: $(a + b)^{n} \leq 2^{n-1}(a^{n}+b^{n}) $ I tried induction theorem, when $n = 1$ it is obviously right. But, say $n=k$, It does not make sense since I cannot ...
1
vote
2answers
43 views

help with (simple?) inequality

Let $x\in(0,1)$ be any and let $0<a<b<\frac{1}{2}$, I need to show that $$1-(1-x^a)(1-x^{1-a})>1-(1-x^b)(1-x^{1-b}).$$ Any suggestions? References? In practice I need to solve a more ...
0
votes
2answers
39 views

Prove a inequality about integral and summation

If $f(x)$ is monotonic increasing on the interval $a\leq x < \infty$, could we prove following inequality formally? \begin{equation} f(a+k) \leq \int_{a+k}^{a+k+1} f(t) dt \leq f(a+k+1) ...