Questions on proving and manipulating inequalities.
33
votes
3answers
601 views
Prove $(\dfrac{2}{5})^{\frac{2}{5}}<\ln{2}$
Inadvertently, I find this interesting inequality,But this problem have nice solution?
prove that
$$\ln{2}>(\dfrac{2}{5})^{\frac{2}{5}}$$
This problem have nice solution? Thank you.
ago,I find ...
32
votes
7answers
1k views
Inequality: $(a^3+a+1)(b^3+b+1)(c^3+c+1) \leq 27$
Let be $a,b,c \geq 0$ such that: $a^2+b^2+c^2=3$.
Prove that:
$$(a^3+a+1)(b^3+b+1)(c^3+c+1) \leq 27.$$
I try to apply $GM \leq AM$ for $x=a^3+a+1$, $y=b^3+b+1,z=c^3+c+1$ and
$$\displaystyle ...
29
votes
4answers
771 views
AM-GM-HM Triplets
I want to understand what values can be simultaneously attained as the arithmetic (AM), geometric (GM), and harmonic (HM) means of finite sequences of positive real numbers. Precisely, for what points ...
29
votes
3answers
1k views
Inequality for expected value
A colleague popped into my office this afternoon
and asked me the following question. He told me there is a
clever proof when $n=2$. I couldn't do
anything with it, so I thought I'd post it here and ...
29
votes
6answers
835 views
A sub-additivity inequality
In trying to understand a result of D. Rider (Trans. AMS, 1973) I've got stuck on a lemma that he uses. At one point he makes a step without comment or explanation, but I can't see why it works.
Here ...
28
votes
6answers
2k views
What is the larger of the two numbers?
What is the larger of the two numbers?
$$\sqrt{2}^{\sqrt{3}} \mbox{ or } \sqrt{3}^{\sqrt{2}}\, \, \; ?$$
I solved this, and I think that is an interesting elementary problem. I want different points ...
27
votes
3answers
843 views
A combinatorial proof of $n^n(n+2)^{n+1}>(n+1)^{2n+1}$?
The statement is, of course, simply that the sequence $\left(1+\frac{1}{n}\right)^n$ is increasing.
Since the numbers $n^m$ have quite natural combinatorial interpretations, it makes me wonder if a ...
23
votes
3answers
835 views
Random Variable Inequality
Doing a little reading over the break (The Probabilistic Method by Alon and Spencer); can't come up with the solution for this seemingly simple (and perhaps even a little surprising?) result:
(A-S ...
22
votes
4answers
1k views
Which is bigger?
In my classes I sometimes have a contest concerning who can write the largest number in ten symbols. It almost never comes up, but I'm torn between two "best" answers: a stack of ten 9's (exponents) ...
21
votes
6answers
1k views
Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $
How could we prove that this inequality holds
$$ \left(1+\frac{1}{n+1}\right)^{n+1} \gt \left(1+\frac{1}{n} \right)^{n} $$
where $n \in \mathbb{N}$, I think we could use the AM-GM inequality ...
21
votes
2answers
378 views
An “AGM-GAM” inequality
For positive real numbers $x_1,x_2,\ldots,x_n$ and any $1\leq r\leq n$ let $A_r$ and $G_r$ be , respectively, the arithmetic mean and geometric mean of $x_1,x_2,\ldots,x_r$.
Is it true that the ...
20
votes
2answers
972 views
I'm not sure about this inequality (how to prove or disprove it?)
For $a_1,...,a_n,b_1,...,b_n>0,\quad$ define $a:=\sum a_i,\ b:=\sum b_i,\ s:=\sum \sqrt{a_ib_i}$.
Is the following inequality true?:
$${\frac{\Bigl(\prod a_i^{a_i}\Bigr)^\frac1a}a \cdot ...
19
votes
1answer
396 views
Computing the best constant in classical Hardy's inequality
Classical Hardy's inequality (cfr. Hardy-Littlewood-Polya Inequalities, Theorem 327)
If $p>1$, $f(x) \ge 0$ and $F(x)=\int_0^xf(y)\, dy$ then
$$\tag{H} \int_0^\infty ...
18
votes
2answers
359 views
Trigonometric Inequality. $\sin{1}+\sin{2}+\ldots+\sin{n} <2$ .
How can I prove the following trigonometric inequality :
$$\sin1+\sin2 +\ldots+\sin n <2$$ with $n \in \mathbb{N}^{*}$.
The problem is that I don't know how to start this problem, I try to ...
17
votes
26answers
3k views
17
votes
8answers
718 views
Comparing $2013!$ and $1007^{2013}$
I have to compare the following two numbers:
$$2013! \text{ and } 1007^{2013}$$
where $n! = 1 \times 2 \times \cdots \times (n-1) \times n$.
I tried in different ways to group the $1 \times 2 ...
17
votes
6answers
828 views
$m!n! < (m+n)!$ Proof?
Prove that if $m$ and $n$ are positive integers then $m!n! < (m+n)!$
Given hint:
$m!= 1\times 2\times 3\times\cdots\times m$ and $1<m+1, 2<m+2, \ldots , n<m+n$
It looks simple but ...
17
votes
3answers
396 views
An Inequality Involving Bell Numbers: $B_n^2 \leq B_{n-1}B_{n+1}$
The following inequality came up while trying to resolve a conjecture about a certain class of partitions (the context is not particularly enlightening):
$$
B_n^2 \leq B_{n-1}B_{n+1}
$$
for $n \geq ...
17
votes
2answers
677 views
Geometric proof for inequality
While on AOPS, I saw this interesting problem. I was wondering how many different approaches could be used to tackle the problem.
In other words I am looking for interesting and unique ways to solve ...
17
votes
1answer
587 views
Do inequalities that hold for infinite sums hold for integrals too?
Let $\mathbb{R}_{\geq0}$ denote the set of non-negative reals and $+\infty$, and $\mathbb{Z}^+$ denote the set of positive integers. I will also let $\lambda$ denote the Lebesgue measure on ...
16
votes
6answers
427 views
$\log_9 71$ or $\log_8 61$
I am trying to know which one is bigger :$$\log_9 71$$ or $$\log_8 61$$ how can i know without using a calculator ?
16
votes
2answers
419 views
Inequality on the side lengths of a triangle: $\left| \frac{a}{b} + \frac{b}{c} + \frac{c}{a} - \frac{a}{c} - \frac{b}{a} - \frac{c}{b} \right| < 1$.
This problem is taken from the Kosovo Mathematical Olympiad for Grade-$ 10 $ students.
Let $ a $, $ b $ and $ c $ be the lengths of the edges of a given triangle.
How can one prove the following ...
15
votes
6answers
831 views
Simple proof that $8\left(\frac{9}{10}\right)^8 > 1$
This question is motivated by a step in the proof given here.
$\begin{align*}
8^{n+1}-1&\gt 8(8^n-1)\gt 8n^8\\
&=(n+1)^8\left(8\left(\frac{n}{n+1}\right)^8\right)\\
&\geq ...
15
votes
4answers
255 views
Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$
Let $a,b,c$ are non-negative numbers, such that $a+b+c = 3$.
Prove that $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$
Here's my idea:
$\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$
...
15
votes
1answer
198 views
$x^3-3x-3=0$, prove that $10^x<127$
$x$ is the real root of the equation $$3x^3-5x+8=0,\tag 1$$ prove that $$e^x>\frac{40}{237}.$$
I find this inequality in a very accidental way,I think it's very difficult,because the actual value ...
14
votes
4answers
605 views
Purely “algebraic” proof of Young's Inequality
Young's inequality states that if $a, b \geq 0$, $p, q > 0$, and $\frac{1}{p} + \frac{1}{q} = 1$, then $$ab\leq \frac{a^p}{p} + \frac{b^q}{q}$$ (with equality only when $a^p = b^q$). Back when I ...
14
votes
2answers
409 views
How to show that $\frac{\pi}{5}\leq\int_0^1 x^x\,dx\leq\frac{\pi}{4}$
Show that:
$$\frac{\pi}{5}\leq\int_0^1 x^x\,dx\leq\frac{\pi}{4}$$
All I've got so far is that the minimum of $x^x$ is $e^{-1/e}$. At this point I could
compare $\pi/5$ to $e^{-1/e}$ but I'm ...
14
votes
4answers
219 views
CSB inquality: is $\|x\|^2\|y\|^2 - \langle x,y \rangle^2$ a square in any obvious way?
Suppose $x=(x_1,x_2),y = (y_1,y_2) \in \mathbb{R}^2$. I noticed that
\begin{align*}
\|x\|^2 \|y\|^2 - \langle x,y \rangle^2 &=
x_1^2y_1^2 + x_1^2 y_2^2 + x_2^2 y_1^2 + x_2^2 y_2 ^2 - (x_1^2 y_1^2 ...
14
votes
1answer
179 views
Prove $\frac{1}{2\sqrt{2}+1}+\frac{1}{3\sqrt{3}+2\sqrt{2}}+\cdots+\frac{1}{100\sqrt{100}+99\sqrt{99}}<\frac{9}{10}$
What would you suggest for the following inequality?
$$\frac{1}{2\sqrt{2}+1}+\frac{1}{3\sqrt{3}+2\sqrt{2}}+\cdots+\frac{1}{100\sqrt{100}+99\sqrt{99}}<\frac{9}{10}$$
Thanks in advance!
Sis.
EDIT: ...
14
votes
2answers
600 views
Sum inequality: $\sum_{k=1}^n \frac{\sin k}{k} \le \pi-1$
I'm interested in finding an elementary proof for the following sum inequality:
$$\sum_{k=1}^n \frac{\sin k}{k} \le \pi-1$$
If this inequality is easy to prove, then one may easily prove that the sum ...
14
votes
4answers
847 views
An Inequality problem relating $\prod\limits^n(1+a_i^2)$ and $\sum\limits^n a_i$
Let $(a_1,\space a_2,\space \cdots, \space a_n) \in \mathbb R^n_+$ such that $\displaystyle \prod^n_{i=1 }a_i = 1$. Prove that $$\displaystyle \prod^n_{i=1} (1+a_i^2) \le \cfrac ...
14
votes
1answer
293 views
How to prove this inequality in Euclidean space?
Prove that
$$\begin{align*}&|a+b||a+c|+|a+b||b+c|+|a+c||b+c|\\
\leq &(|a|+|b|+|c|) \cdot |a+b+c|+|a||b|+|a||c|+|b||c|\end{align*}$$
in Euclidean space $\mathbb{R}^n$.
I have been ...
14
votes
1answer
155 views
How to prove $\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $?
I want to prove the inequality
$$\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $$
There are some obstacles I face: the indefinite integral cannot be expressed in terms of ...
14
votes
1answer
243 views
Trace inequality for real matrices
Is there any general result characterizing real matrices $A$ such that
$$[\mathrm{tr}(A)]^2\leq n\mathrm{tr}(A^2)?$$
I can see that the inequality holds if:
all eigenvalues of $A$ are real (by the ...
14
votes
3answers
345 views
Inequality for cosines
Is the following inequality in a triangle known?
$$4(\cos A + \cos B + \cos C) \le 3 + \cos \left(\frac{B-C}{2}\right) + \cos \left(\frac{C-A}{2}\right) + \cos \left(\frac{A-B}{2}\right)$$
It looks ...
13
votes
3answers
778 views
Stuck trying to prove an inequality
I have been trying to prove (the left half of) the following inequality:
$$ \underbrace{\sum_i \sum_j |x_i| \le \sum_i \sum_j |x_i + x_j|}_\textrm{?} \le 2 \sum_i \sum_j |x_i|$$
(All $x_i$s are ...
13
votes
4answers
438 views
Showing that $ |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$
For every nonnegative integer $n$ and every real number $ x$ prove the inequality:
$$\sum_{k=0}^n|\cos(2^kx)|= |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$$
13
votes
1answer
281 views
On the equality case of the Hölder and Minkowski inequalites
I'm following the book Measure and Integral of Richard L. Wheeden and Antoni Zygmund. This is the problem 4 of chapter 8.
Consider $E\subseteq \mathbb{R}^n$ a measurable set. In the following all the ...
13
votes
3answers
411 views
Proving that $\sum_{i=0}^{n}\binom{n}{i}i^{n-i}(n-i)^{i}\le\frac{1}{2}n^n$
How can we prove that
$$\displaystyle\sum_{i=0}^{n}\binom{n}{i}i^{n-i}(n-i)^{i}\le\dfrac{1}{2}n^n$$
where $\displaystyle\binom{n}{i}=\dfrac{n!}{i!(n-i)!}$.
This inequality is very interesting. I ...
13
votes
1answer
366 views
Proving a complicated inequality involving integers
Let $a,b,c,d$ be integers such that $$\left( \begin{matrix} a & b \\ c & d \end{matrix} \right) = \left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right) \mod 2$$ $$ ad-bc =1$$ ...
12
votes
6answers
808 views
Proving the inequality $e^{-2x}\leq 1-x$
How do I prove the inequality $e^{-2x}\leq1-x$ for $0\leq x\leq1/2$?
12
votes
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 $b^2 + c^2 \geq 2bc~$.
...
12
votes
3answers
592 views
Motivation for triangle inequality
Triangle inequality is used in one context or the other in analysis.
To list a few
$$ \|x+y\| \leq \|x\| + \|y\| $$
$$ d(x,y) \leq d(x,z) + d(z,y) $$
$$ \mu(A \cup B) \leq \mu(A) + \mu(B) $$
What ...
12
votes
2answers
417 views
Complex-number inequality $| z_1 z_2 \ldots z_m - 1 | \leq e^{|z_1 - 1| + \ldots + |z_m - 1|} - 1$
Let $z_1, z_2 \ldots z_m$ be complex numbers, $m \in \mathbb{N}$. Can anybody tell me how to prove the following inequality?
$| z_1 z_2 \ldots z_m - 1 | \leq e^{|z_1 - 1| + \ldots + |z_m - 1|} - 1$
...
12
votes
2answers
527 views
the least value for :$\frac{a}{b^3+54}+\frac{b}{c^3+54}+\frac{c}{a^3+54}$
For every $a,b,c$ non-negative real number such that:$a+b+c=1$ how to find the least value for :
$$\frac{a}{b^3+54}+\frac{b}{c^3+54}+\frac{c}{a^3+54}$$
12
votes
1answer
245 views
Chebyshev: Proof $\prod \limits_{p \leq 2k}{\;} p > 2^k$
How do I prove the following:
$$\prod_{p \leq 2k} \; p > 2^k \text{ with } p \in \mathbb{P}$$
I tried induction, but I didn't know how to go on because I don't have a look at all numbers.
...
12
votes
2answers
464 views
Inequality. $\sqrt{\frac{11a}{5a+6b}}+\sqrt{\frac{11b}{5b+6c}}+\sqrt{\frac{11c}{5c+6a}} \leq 3$
Let $a,b,c$ be positive numbers . Prove the following inequality:
$$\sqrt{\frac{11a}{5a+6b}}+\sqrt{\frac{11b}{5b+6c}}+\sqrt{\frac{11c}{5c+6a}} \leq 3.$$
What I tried:
I used ...
12
votes
2answers
118 views
Proving the inequality $\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin(1/k^2)}{\cos^2 (1/(k+1))}$
How am I supposed to prove this inequality?
$$\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin\left(\frac{1}{k^2}\right)}{\cos^2 \left(\frac{1}{k+1}\right)}$$
Jordan inequality might be an option but led me ...
12
votes
1answer
489 views
Combinatorial proof of arithmetic geometric mean inequality
It is a well known fact that for positive reals $x_1, x_2, \dots, x_n$, their arithmetic mean is no less than their geometric mean:
$$ \frac{x_1 + x_2 + \dots + x_n}{n} \ge \sqrt[n]{x_1 x_2 \dots ...
12
votes
1answer
234 views
Proving $\pi(\frac1A+\frac1B+\frac1C)\ge(\sin\frac A2+\sin\frac B2+\sin\frac C2)(\frac 1{\sin\frac A2}+\frac 1{\sin\frac B2}+\frac 1{\sin\frac C2})$
Let $\Delta ABC$, prove that
$$\pi\left(\dfrac{1}{A}+\dfrac{1}{B}+\dfrac{1}{C}\right)\ge \left(\sin{\dfrac{A}{2}}+\sin{\dfrac{B}{2}}+\sin{\dfrac{C}{2}} \right) ...
