Tagged Questions

Questions on proving and manipulating inequalities.

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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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) ...
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 ...
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 ...
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 ...
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 ...
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 ...
26answers
3k views

How can I prove that $xy\leq x^2+y^2$?

How can I prove that $xy\leq x^2+y^2$?
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 ...
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 ...
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 ...
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 ...
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 ...
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 ?
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 ...
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 ...
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: $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ ...
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 ...
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 ...
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 ...
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 ...
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: ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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: $$\sum_{k=0}^n|\cos(2^kx)|= |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$$
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 ...
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 ...
1answer
366 views

Proving a complicated inequality involving integers

Let $a,b,c,d$ be integers such that $$\left( \begin{matrix} a & b \\ c & d \end{matrix} \right) = \left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right) \mod 2$$ $$ad-bc =1$$ ...
6answers
808 views

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

How do I prove the inequality $e^{-2x}\leq1-x$ for $0\leq x\leq1/2$?
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 $b^2 + c^2 \geq 2bc~$. ...
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 ...
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? $| z_1 z_2 \ldots z_m - 1 | \leq e^{|z_1 - 1| + \ldots + |z_m - 1|} - 1$ ...
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 : $$\frac{a}{b^3+54}+\frac{b}{c^3+54}+\frac{c}{a^3+54}$$
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. ...
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 ...
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 ...
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 ...
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) ...
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