Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities. Do not use this tag just because an inequality appears somewhere in your question.

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How does one prove the determinant inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Let $\,A,B,C\in M_{n}(\mathbb C)\,$ be Hermitian and positive definite matrices such that $A+B+C=I_{n}$, where $I_{n}$ is the identity matrix. Show that $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n \...
math110's user avatar
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144 votes
10 answers
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Is the blue area greater than the red area?

Problem: A vertex of one square is pegged to the centre of an identical square, and the overlapping area is blue. One of the squares is then rotated about the vertex and the resulting overlap is red. ...
Mr Pie's user avatar
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131 votes
27 answers
51k views

Proofs of AM-GM inequality

The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual ...
Michiel Van Couwenberghe's user avatar
129 votes
14 answers
30k views

Olympiad Inequality $\sum\limits_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$

$x,y,z >0$, prove $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$ Note: Often Stack Exchange asked to show some work before answering the ...
HN_NH's user avatar
  • 4,351
125 votes
7 answers
216k views

Prove that $||x|-|y||\le |x-y|$

I've seen the full proof of the Triangle Inequality \begin{equation*} |x+y|\le|x|+|y|. \end{equation*} However, I haven't seen the proof of the reverse triangle inequality: \begin{equation*} ||x|-|...
Anonymous's user avatar
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119 votes
26 answers
80k views

Simplest or nicest proof that $1+x \le e^x$

The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or ...
Ashley Montanaro's user avatar
113 votes
8 answers
80k views

Much less than, what does that mean?

What exactly does $\ll$ mean? I am familiar that this symbol means much less than. ...but what exactly does "much less than" mean? (Or the corollary, $\gg$) On Wikipedia, the example they use is ...
user avatar
111 votes
1 answer
5k views

Is this continuous analogue to the AM–GM inequality true?

First let us remind ourselves of the statement of the AM–GM inequality: Theorem: (AM–GM Inequality) For any sequence $(x_n)$ of $N\geqslant 1$ non-negative real numbers, we have $$\frac1N\sum_k x_k ...
user1892304's user avatar
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108 votes
15 answers
98k views

How to prove that exponential grows faster than polynomial?

In other words, how to prove: For all real constants $a$ and $b$ such that $a > 1$, $$\lim_{n\to\infty}\frac{n^b}{a^n} = 0$$ I know the definition of limit but I feel that it's not enough to ...
faceclean's user avatar
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106 votes
15 answers
16k views

Comparing $\pi^e$ and $e^\pi$ without calculating them

How can I compare (without calculator or similar device) the values of $\pi^e$ and $e^\pi$ ?
Mirzodaler's user avatar
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98 votes
8 answers
13k views

Prove $\left(\frac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$

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 ...
math110's user avatar
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95 votes
2 answers
6k views

Prove elementarily that $\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}$ is strictly decreasing

Prove without calculus that the sequence $$L_{n}=\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}, \space n\in \mathbb N$$ is strictly decreasing.
user 1591719's user avatar
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89 votes
7 answers
11k views

If $a+b=1$ then $a^{4b^2}+b^{4a^2}\leq1$

Let $a$ and $b$ be positive numbers such that $a+b=1$. Prove that: $$a^{4b^2}+b^{4a^2}\leq1$$ I think this inequality is very interesting because the equality "occurs" for $a=b=\frac{1}{2}$ ...
Michael Rozenberg's user avatar
85 votes
3 answers
14k views

Fastest way to check if $x^y > y^x$?

What is the fastest way to check if $x^y > y^x$ if I were writing a computer program to do that? The issue is that $x$ and $y$ can be very large.
learner's user avatar
  • 905
77 votes
11 answers
235k views

Proof of triangle inequality

I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :)
ivan's user avatar
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73 votes
8 answers
7k views

How can one prove that $e<\pi$?

This question is inspired by another one, asking to prove that something approximately equal to $1.2$ is bigger than something approximately equal to $0.9$. The numerical answer to this question was (...
Start wearing purple's user avatar
68 votes
4 answers
69k views

Intuition behind logarithm inequality: $1 - \frac1x \leq \log x \leq x-1$

One of fundamental inequalities on logarithm is: $$ 1 - \frac1x \leq \log x \leq x-1 \quad\text{for all $x > 0$},$$ which you may prefer write in the form of $$ \frac{x}{1+x} \leq \log{(1+x)} \leq ...
Federico Magallanez's user avatar
67 votes
29 answers
27k views

How to explain for my daughter that $\frac {2}{3}$ is greater than $\frac {3}{5}$?

I was really upset while I was trying to explain for my daughter that $\frac 23$ is greater than $\frac 35$ and she always claimed that $(3$ is greater than $2$ and $5$ is greater than $3)$ then $\...
Akam's user avatar
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67 votes
9 answers
39k views

How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?

Let $A$ and $B$ be two matrices which can be multiplied. Then $$\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B)).$$ I proved $\operatorname{rank}(AB) \...
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66 votes
6 answers
23k views

How do you show monotonicity of the $\ell^p$ norms?

I can't seem to work out the inequality $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$ for $p \leq q$ (which I'm assuming is the way to go about it).
user1736's user avatar
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65 votes
8 answers
6k views

What is Cauchy Schwarz in 8th grade terms?

I'm an 8th grader. After browsing aops.com, a math contest website, I've seen a lot of problems solved by Cauchy Schwarz. I'm only in geometry (have not started learning trigonometry yet). So can ...
Schneider's user avatar
  • 749
63 votes
18 answers
10k views

Which is larger? $20!$ or $2^{40}$?

Someone asked me this question, and it bothers the hell out of me that I can't prove either way. I've sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of ...
Alec's user avatar
  • 4,094
61 votes
3 answers
12k views

A very general method for proving inequalities. Too good to be true?

Update I 'repaired' this method, but it changed a lot and I have some different questions, so I posted it separately here. As training for the olympiad, I have to solve a lot of inequalities. ...
Mastrem's user avatar
  • 8,266
60 votes
13 answers
16k views

Why does $\frac{1}{x} < 4$ have two answers?

Solving $\frac{1}{x} < 4$ gives me $x > \frac{1}{4}$. The book however states the answer is: $x < 0$ or $x > \frac{1}{4}$. My questions are: Why does this inequality has two answers (...
Augusto Dias Noronha's user avatar
60 votes
6 answers
13k views

Why does the Cauchy-Schwarz Inequality even have a name?

When I came across the Cauchy-Schwarz inequality the other day, I found it really weird that this was its own thing, and it had lines upon lines of proof. I've always thought the geometric definition ...
Second Wind's user avatar
  • 1,059
60 votes
2 answers
2k views

How prove this inequality $\sin{\sin{\sin{\sin{x}}}}\le\frac{4}{5}\cos{\cos{\cos{\cos{x}}}}$

Nice Question: let $x\in [0,2\pi]$, show that: $$\sin{\sin{\sin{\sin{x}}}}\le\dfrac{4}{5}\cos{\cos{\cos{\cos{x}}}}?$$ I know this follow famous problem(1995 Russia Mathematical olympiad) $$\...
math110's user avatar
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59 votes
14 answers
23k views

How to prove that $\log(x)<x$ when $x>1$? [duplicate]

It's very basic but I'm having trouble to find a way to prove this inequality $\log(x)<x$ when $x>1$ ($\log(x)$ is the natural logarithm) I can think about the two graphs but I can't find ...
Gianolepo's user avatar
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59 votes
1 answer
109k views

Is there a "greater than about" symbol?

To indicate approximate equality, one can use ≃, ≅, ~, ♎, or ≒. I need to indicate an approximate inequality. Specifically, I know A is greater than a quantity of approximately B. Is there a way to ...
foobarbecue's user avatar
55 votes
14 answers
5k views

Simplest way to get the lower bound $\pi > 3.14$

Inspired from this answer and my comment to it, I seek alternative ways to establish $\pi>3.14$. The goal is to achieve simpler/easy to understand approaches as well as to minimize the calculations ...
Paramanand Singh's user avatar
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55 votes
4 answers
2k views

Prove/disprove $(\int_0^{2 \pi} \!\!\cos f(x) \, d x)^2+(\int_0^{2 \pi}\!\!\! \sqrt{(f'(x))^2+\sin ^2 f(x)} \, dx)^2\ge 4\pi^2$

Let $f(x)$ be a differentiable function on $[0,2\pi]$ s.t. $0\leq f(x)\leq 2\pi$ and $f(0)=f(2\pi)$. Prove or disprove that $$ \left(\int_0^{2 \pi} \cos f(x) \,d x\right)^2+\left(\int_0^{2 \pi} \sqrt{(...
FFjet's user avatar
  • 5,041
54 votes
3 answers
4k 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 > ...
J. M. ain't a mathematician's user avatar
52 votes
11 answers
24k views

Which of the numbers $1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$ is largest, and how to find out without calculator? [closed]

$1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$. I got this question in an Application of Derivatives test. I think log might be used here to compare the values, but even then the values ...
anshabhi's user avatar
  • 751
52 votes
4 answers
23k 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 ...
asmeurer's user avatar
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52 votes
5 answers
3k 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 ...
Srivatsan's user avatar
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51 votes
0 answers
2k views

A Nice Problem In Additive Number Theory

$\color{red}{\mathbf{Problem\!:}}$ $n\geq3$ is a given positive integer, and $a_1 ,a_2, a_3, \ldots ,a_n$ are all given integers that aren't multiples of $n$ and $a_1 + \cdots + a_n$ is also not a ...
ShBh's user avatar
  • 6,034
50 votes
5 answers
49k views

Prove $\limsup\limits_{n \to \infty} (a_n+b_n) \le \limsup\limits_{n \to \infty} a_n + \limsup\limits_{n \to \infty} b_n$

I am stuck with the following problem. Prove that $$\limsup_{n \to \infty} (a_n+b_n) \le \limsup_{n \to \infty} a_n + \limsup_{n \to \infty} b_n$$ I was thinking of using the triangle inequality ...
hyg17's user avatar
  • 5,017
50 votes
3 answers
23k views

On the equality case of the Hölder and Minkowski inequalities

I'm following the book Measure and Integral of Richard L. Wheeden and Antoni Zygmund. This is problem 4 of chapter 8. Consider $E\subseteq \mathbb{R}^n$ a measurable set. In the following all the ...
leo's user avatar
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49 votes
7 answers
4k views

Inequality from Chapter 5 of the book *How to Think Like a Mathematician*

This is from the book How to think like a Mathematician, How can I prove the inequality $$\sqrt[\large 7]{7!} < \sqrt[\large 8]{8!}$$ without complicated calculus? I tried and finally obtained ...
Gwydyon's user avatar
  • 591
47 votes
6 answers
7k views

What is the intuition behind Chebyshev's Inequality in Measure Theory

Chebyshev's Inequality Let $f$ be a nonnegative measurable function on $E .$ Then for any $\lambda>0$, $$ m\{x \in E \mid f(x) \geq \lambda\} \leq \frac{1}{\lambda} \cdot \int_{E} f. $$ What ...
Bill's user avatar
  • 4,347
47 votes
5 answers
3k views

For $a$, $b$, $c$, $d$ the sides of a quadrilateral, show $ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$. (A generalization of IMO 1983 problem 6)

Let $a$, $b$, $c$, and $d$ be the lengths of the sides of a quadrilateral. Show that $$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0 \tag{$\star$}$$ Background: The well known 1983 IMO Problem 6 is ...
math110's user avatar
  • 93.1k
46 votes
6 answers
43k views

Sylvester rank inequality: $\operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n$ [duplicate]

If $A$ and $B$ are two matrices of the same order $n$, then $$ \operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n. $$ I don't know how to start proving this inequality. I ...
Ben Ward's user avatar
  • 665
46 votes
2 answers
4k views

Prove that $a\sqrt{a^2+bc}+b\sqrt{b^2+ac}+c\sqrt{c^2+ab}\geq\sqrt{2(a^2+b^2+c^2)(ab+ac+bc)}$

Let $a$, $b$ and $c$ be non-negative numbers. Prove that: $$a\sqrt{a^2+bc}+b\sqrt{b^2+ac}+c\sqrt{c^2+ab}\geq\sqrt{2(a^2+b^2+c^2)(ab+ac+bc)}.$$ I have a proof, but my proof is very ugly: it's ...
Michael Rozenberg's user avatar
46 votes
4 answers
2k views

Is each of $\int_0^\infty\frac{dx}{x^x},\int_0^\infty\frac{dx}{x^{x^{x^x}}},\int_0^\infty\frac{dx}{x^{x^{x^{x^{x^x}}}}},\cdots$ less than $2$?

A few years ago I asked about the inequality Prove that $\int_0^\infty\frac1{x^x}\, dx<2$. As I came back to revisit it, I found that each of the following tetration integrals $$\int_0^\infty\frac{...
TheSimpliFire's user avatar
  • 27k
46 votes
1 answer
7k views

A proof of the Isoperimetric Inequality - how does it work?

Here is a nice proof of the isoperimetric inequality (equality part ommited): Isoperimetric Inequality If $\gamma$ is any simple closed piecewise $C^1$ curve of length $l$, with it's interior having ...
milcak's user avatar
  • 4,129
45 votes
4 answers
8k views

cosh x inequality

While reading an article on Hoeffding's Inequality, I came across a curious inequality. Namely $$\cosh x \leq e^{x^2/2} \quad \forall x \in \mathbb{R}$$ I tried many ways to prove it and finally, ...
Gautam Shenoy's user avatar
44 votes
14 answers
51k views

Proofs of the Cauchy-Schwarz Inequality?

How many proofs of the Cauchy-Schwarz inequality are there? Is there some kind of reference that lists all of these proofs?
44 votes
1 answer
47k views

Integral Inequality Absolute Value: $\left| \int_{a}^{b} f(x) g(x) \ dx \right| \leq \int_{a}^{b} |f(x)|\cdot |g(x)| \ dx$

Suppose we are given the following: $$\left| \int_{a}^{b} f(x) g(x) \ dx \right| \leq \int_{a}^{b} |f(x)|\cdot |g(x)| \ dx$$ How would we prove this? Does this follow from Cauchy Schwarz? Intuitively ...
fourierguy's user avatar
43 votes
2 answers
32k views

Proof of upper-tail inequality for standard normal distribution

$X \sim \mathcal{N}(0,1)$, then to show that for $x > 0$, $$ \mathbb{P}(X>x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \>. $$
user avatar
43 votes
8 answers
4k 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 \...
43 votes
1 answer
1k views

Stronger version of AMM problem 11145 (April 2005)?

How to show that for $a_1,a_2,\cdots,a_n >0$ real numbers and for $n \ge 3$: $$\sum_{k=1}^{n}\dfrac{k}{a_{1}+a_{2}+\cdots+a_{k}}\le\left(2-\dfrac{7\ln{2}}{8\ln{n}}\right)\sum_{k=1}^{n}\dfrac{1}{...
r9m's user avatar
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