Questions on proving and manipulating inequalities.

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3answers
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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 ...
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3answers
487 views

Another symmetric inequality

How would one show that for positive $a,b,c,d$ and $a+b+c+d = 4$ that $$ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \leq \frac{4}{abcd} $$
4
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1answer
1k views

Bounds of Euler's totient function?

Conjecture : Let $\phi(m)$ be Euler's totient function $1 \leq \phi(m) \leq \lceil \frac{m-1}{2} \rceil ~~$ if $~~m~~$ is even $\lceil \frac{m+1}{3} \rceil \leq\phi(m) \leq m-1 ~~$ ...
2
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1answer
175 views

A symmetric inequality [duplicate]

Possible Duplicate: Cauchy-Schwarz inequality and three-letter identities (exercise 1.4 from “The Cauchy-Schwarz Master Class”) Is it true for all $x, y, z > 0$ that $$ x + ...
4
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1answer
1k views

Proving the Schwarz Inequality for Complex Numbers using Induction

I want to prove the following version of the Schwarz Inequality for complex numbers $a_1, a_2, \ldots, a_n \in \mathbb{C}$ and $b_1, b_2, \ldots, b_n \in \mathbb{C}$: $$|\sum_{j=1}^n a_j ...
0
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4answers
198 views

Proof that $\cos x \leq \frac{\sin x}{x}, x \in [0,\pi ]$

As the title implies I am trying to prove the inequality $$\cos x \leq \frac{\sin x}{x}, x \in [0,\pi ]$$ However, I am not sure how to approach it. Any help would be greatly appreciated.
2
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1answer
267 views

Rudin's assertion that if $t = x/(1 + x)$ then $0 \leq t < 1$

I'm having trouble understanding one step in the proof of Theorem 1.21 in Rudin's Principles of Mathematical Analysis. Theorem 1.21 For every real $x > 0$ and every integer $n > 0$ there is one ...
2
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2answers
127 views

Prove: $\left| \dfrac{a}{|a|^{\alpha}} - \dfrac{b}{|b|^{\alpha}} \right| \leq 2^{\alpha}|a - b|^{1-\alpha}$

I believe the following inequality is true. Can anyone prove it? Let $a,b \in \mathbb{R}$ with $a,b \neq 0$ and $0 \leq \alpha < 1$, then $$ \left| \dfrac{a}{|a|^{\alpha}} - ...
9
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1answer
856 views

Olympiad Style Inequality

I don't quite remember where this problem is from. I came across is sometime last summer, when I was in an olympiad-problem mood and I decided to improve my inequality skills. Suppose $a,b,c > 0$. ...
3
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2answers
932 views

The inequality $b^n - a^n < (b - a)nb^{n-1}$

I'm trying to figure out why $b^n - a^n < (b - a)nb^{n-1}$. Using just algebra, we can calculate $ (b - a)(b^{n-1} + b^{n-2}a + \ldots + ba^{n-2} + a^{n-1}) $ $ = (b^n + b^{n-1}a + \ldots + ...
4
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1answer
307 views

How to prove $f(x)>g(x)$ by showing $f'(x)>g'(x)$ for some $x>x_0$

Is it possible to show that some function $f(x)$ is larger than $g(x)$ if I can show that its derivative is larger for some $x>x_0$? As an example I can think of $f(x)=(1+x)^n , \ g(x)=1+nx+ ...
1
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1answer
191 views

unit ball question

If $1\leq p<q$, show that the unit ball $l_{n}^p(\mathbb{R})$ is contained in the unit ball $l_{n}^q(\mathbb{R})$. Well the definition of $l_{n}^p(\mathbb{R})$ is that for ...
0
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1answer
137 views

Inequality regarding exponents(similar to Bernoulli's inequality)

I was trying to solve a problem today when I felt that if the following is proved, we are done: $(1+a)^y<(1+ay)$ for $0<y<1$ for all non-zero reals a,y.I am not able to make much progress on ...
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8answers
2k views

Which of the numbers $99^{100}$ and $100^{99}$ is the larger one?

Which of the numbers $99^{100}$ & $100^{99}$ is the larger? Solve without using logarithms.
8
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4answers
3k views

Greatest prime factor of $n$ is less than square root of $n$, proof

I remember reading this somewhere but I cannot locate the proof.
7
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1answer
310 views

Maximum subset sum of $d$-dimensional vectors

This is a $d$-dimensional generalisation of the post Inequality with Complex Numbers. (See my comment under Robert Israel's answer.) Generalising Potato's proof for $d$-dimensions, we can show the ...
2
votes
1answer
521 views

How to manipulate absolute values when shifting parts in an inequality

I have the following inequality... $|4x - 2| \le 0.5$ I want to manipulate this so it is just $|x|$ on one side, and everything else on the other, but I'm not sure how the absolute value complicates ...
2
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0answers
126 views

another inequality involving complex numbers.

Let $\{z_i\}$, $i=1,2,\ldots,n$ be a set of complex numbers. Then I know that there is a set $J$ such that $$\left|\sum_{j\in J} z_j\right|\ge \frac{1}{\pi} \sum_{k=1}^n |z_k|. $$ However, how do I ...
4
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1answer
218 views

An integral operator inequality

This problem is from exercise 30 on page 196 of Folland's "Real Analysis". Let $K$ be a non-negative measurable function on $[0,\infty)$ and let $\phi(s) = \displaystyle\int_0 ^\infty K(x)x^{s-1} ...
11
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3answers
1k views

Lower bound for $\phi(n)$: Is $n/5 < \phi (n) < n$ for all $n > 1$?

Is it true that : $\frac {n}{5} < \phi (n) < n$ for all $n > 1$ where $\phi (n)$ is Euler's totient function . Since $\phi(n)$ has maximum value when $n$ is a prime it follows that ...
3
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1answer
85 views

An inequality with radicals

If $s_{1}\ge t_{1}\ge t_{2}\ge s_{2}\ge0$, does one always have $(s_{1}-t_{1}+s_{2}+t_{2})^{1/2}\ge\sqrt{s_{1}}-\sqrt{t_{1}}+\sqrt{t_{2}}-\sqrt{s_{2}}$? Thanks a lot!
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3answers
102 views

finding bound for the integral

I am trying to get bound for the following integral $$ \int_0^{\infty}\frac{1}{|x|^r}dx, \mbox{for } 1\leq r< \infty $$ In particular, the bound of the form $\frac{constant}{r}$. Sorry, we can ...
4
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1answer
67 views

Connection between some inequalities

Let $I \subset \mathbb{R}$ be an interval, $f: I \rightarrow \mathbb{R}$ be a function and let $n \geq 2, n \in \mathbb{N}$ be fixed number. Let's consider the following conditions: $\displaystyle ...
4
votes
1answer
434 views

Limit of a function satisfying an inequality

If $f(x)+f(y)\leq f(x+y)$ and $f:\mathbb{R}\to\mathbb{R}$, then can we find $\lim_{x\to 0} \frac {f(x)}{x}$? I am not sure whether the question is correct.Thank you.(I tried this idea: ...
2
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1answer
265 views

Moment and tail probability inequality

How can I prove the following inequality: If $r>p$, then \begin{equation} \left\|X\right\|_{p}\le\left(\frac{r}{r-p}\right)^{1/p}\left\|X\right\| _{r,\infty}, \end{equation} where ...
2
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1answer
198 views

Tail inequality using probability generating function

for example there is a Binomial RV $S_n \sim \mathrm{Binomial}(n,\frac{1}{n})$. I'd like to use Markov inequality to find the tail probability: $$ P(S_n \geq \varepsilon) \leq ...
2
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2answers
130 views

That $|a|\leq|b|$ implies existence of complex $z$ satisfying $|z-a|+|z+a|=2|b|$?

I'm looking at the equation $|z-a|+|z+a|=2|b|$. If there are complex values $z$ satisfying this equation, then $$ 2|b|=|z-a|+|z+a|=|a-z|+|z+a|\geq|(a-z)+(z+a)|=|2a|=2|a| $$ so $|a|\leq |b|$. ...
1
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1answer
549 views

Inductive proof of Cauchy's inequality for complex numbers?

I'm trying to put together an inductive proof of Cauchy's inequality for the complex case, $$ \left|\sum_{i=1}^na_ib_i\right|^2\leq\sum_{i=1}^n|a_i|^2\sum_{i=1}^n|b_i|^2. $$ The base case is easy, ...
3
votes
2answers
82 views

$x_n^2 > 2,$ how to show that $x_{n+1}^2 > 2$?

$x_n^2 > 2,$ how to show that $x_{n+1}^2 > 2$? I have tried using induction on this but haven't been able to solve this for a while. The sequence is defined as $x_1 = 2,$ $x_{n+1} = ...
1
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1answer
288 views

Maximum inequality

Can anyone show that ...
2
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1answer
682 views

Hardy's Inequality (corrected)

Reading the questions in this forum, I was interested by the Classical Hardy's inequality: $$\int_0^{\infty}\left(\frac{1}{x}\int_0^xf(s)ds\right)^p dx\leq ...
1
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1answer
708 views

Solving modular inequalities/constraint solving

A few of my current programming problems boil down to solving inequalities over modular domains and possibility could benifit from knowledge of efficient maths/algorithms rather than brute force ...
3
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1answer
344 views

Hardy's inequality again

How can I prove that the constant in classical Hardy's inequality is optimal? $$\int_0^{\infty}\left(\frac{1}{x}\int_0^xf(s)ds\right)^p dx\leq \left(\frac{p}{p-1}\right)^p\int_0^{\infty}(f(x))^pdx,$$ ...
2
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5answers
380 views

Why $x<\tan{x}$ while $0<x<\frac{\pi}{2}$?

In proof of $\displaystyle\lim_{x\rightarrow0}\frac{\sin{x}}{x}=1$ is assumed that $\sin{x}\leq{x}\leq\tan{x}$ while $0<x<\frac{\pi}{2}$. First comparison is clear, arc length must be greater ...
4
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1answer
161 views

Example of function satisfying for fixed $t\in (0,1)$ inequality $f(tx+(1-t)y) \leq tf(x)+(1-t)f(y)$

I would like to know an example of function $f: \mathbb{R} \rightarrow \mathbb{R}$ which is not convex but satisfies for fixed $t\in (0,1)$ the following inequality: $$f(tx+(1-t)y) \leq t ...
1
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3answers
102 views

how can I prove that when $3x+2y\leq 5$, then $x > 1$ implies $y < 1 $?

Basically what I have is that $x,y \in \mathbb{R}$ and that $3x + 2y \leq 5 $, so what I need to prove is that $x > 1 \rightarrow y < 1$ How would you prove this? In some way i know that if I ...
5
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2answers
401 views

A version of Hardy's inequality involving reciprocals.

How can one prove for any sequence of positive numbers $a_n, n\ge1,$ we have $$\sum_{n=1}^\infty \frac{n}{a_1+a_2+a_3+\cdots+a_n}\le 2\sum_{n=1}^\infty \frac{1}{a_n}$$ Added later: Apparently, ...
16
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1answer
554 views

How do you prove Gautschi's inequality for the gamma function?

A few answers here on math.SE have used as an intermediate step the following inequality that is due to Walter Gautschi: $$x^{1-s} < \frac{\Gamma(x+1)}{\Gamma(x+s)} < (x+1)^{1-s},\qquad x > ...
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3answers
1k views

What does this “double less than or equals to” sign mean?

I found this in a Computer Science pseudocode context (see page 4 of this paper).
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1answer
252 views

Inequality for modulus

Let $a$ and $b$ be complex numbers with modulus $< 1$. How can I prove that $\left | \frac{a-b}{1-\bar{a}b} \right |<1$ ? Thank you
2
votes
3answers
133 views

Is this proof about $a^3>a \rightarrow a^5>a $ correct?

Basically what I need is to know if this proof is correct what I need to prove is: if $a^3 > a $ then $a^5>a$ so, what i did was this: $a^3 a^2 > a a^2$ $a^5 > a^3$ because $a^5>a^3$ I ...
4
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1answer
291 views

An inequality involving binomial coefficients

For $k\leq n$, how do I prove that ${2n \choose n}(1-\frac{k}{n})^{k}\leq{2n \choose n+k}$?
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1answer
76 views

verify an inequality

Given $0 < \mu_1<\gamma_1<\mu_2<\gamma_2$, I would like to show $$ ...
12
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6answers
1k views

Proving the inequality $e^{-2x}\leq 1-x$

How do I prove the inequality $e^{-2x}\leq1-x$ for $0\leq x\leq1/2$?
3
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2answers
232 views

Graphing Inequalities in Two Variables

When we have to graph an inequality in two variables, we usually graph the corresponding equality, i.e. the straight line on the coordinate plane, which divides the plane into two parts. Then, we use ...
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0answers
183 views

Proof of Karamata Inequality/Hardy-Littlewood Inequality.

Can anyone provide the proof for the Karamata Inequality?
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2answers
290 views

Determining when an equality holds, based on proven inequality.

I am working on the same question as a poster here and I was able to prove the inequality by re-factoring to: $$ (x-z)^2 + 4y(2-x-z) + 4y^2 \geq 0$$ and arguing that given the conditions, this holds ...
8
votes
1answer
324 views

Derivation of bound on expression involving binomial coefficient from Erdős and Rényi 1959

I'm in the process of working through Erdős and Rényi's 1959 article "On Random Graphs I". In the proof of the first Lemma, equation 14 gives a bound on an expression involving several binomial ...
1
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3answers
66 views

Show $|x|<|x^2+b|$ for real $x$ and $b>\frac{1}{4}$

How do I show $|x|<|x^2+b|$ for real $x$ and $b>\frac{1}{4}$ ? I mean it's clear for $x=0$ and $|x|>1$ but what is if $0<|x|<1$? Please help!
2
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2answers
156 views

Proving Asymptotic Equipartition Property for Gaussian r.v.'s using the Chernoff Bound

I just learned about Chernoff Bounds and am wondering if one can prove the Asymptotic Equipartition Property using them instead of the Weak Law of Large Numbers (which is the consequence of the ...