Questions on proving, manipulating and applying inequalities.

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126
votes
1answer
6k views

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

Question: let matrices $A,B,C\in M_{n}(C)$ be Hermitian and Positive definite matrices, such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ ...
73
votes
6answers
3k views

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

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 ...
60
votes
7answers
4k 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 ...
59
votes
8answers
5k 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 ...
53
votes
2answers
3k 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.
51
votes
2answers
1k 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) ...
48
votes
21answers
4k 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 ...
48
votes
9answers
4k views

Comparing $\pi^{e}$ and $e^{\pi}$

How can I calculate without calculator or something like this the values of $\pi^{e}$ and $e^{\pi}$ in order to compare them ?
48
votes
6answers
8k 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 ...
44
votes
5answers
3k 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 ...
44
votes
4answers
1k 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 ...
43
votes
2answers
992 views

Possible distinct positive real $x,y,z \neq 1$ with $x^{(y^z)} = y^{(z^x)} = z^{(x^y)}$ in cyclic permutation?

Can we have distinct positive real $x,y,z \neq 1$ with $$ x^{\left( y^z \right)} = y^{\left( z^x \right)} = z^{\left( x^y \right)} $$ in cyclic permutation? It does not work well if any ...
42
votes
11answers
4k 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?

$1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$. 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 ...
42
votes
13answers
3k 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 ...
42
votes
7answers
2k 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 ...
41
votes
7answers
2k 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 ...
39
votes
3answers
2k views

A generalization of IMO 1983 problem 6

Note: This question has a bounty that will expire in just a few days. Let $a,b,c$ and $d$ be the lengths of the sides of a quadrilateral. Show that $$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$$ ...
38
votes
2answers
1k views

Prove this inequality with $xyz\le 1$

if $x,y,z>0$ and $\color{red}{xyz\le 1}$, show that $$\color{blue}{\dfrac{x^2-x+1}{x^2+y^2+1}+\dfrac{y^2-y+1}{y^2+z^2+1} +\dfrac{z^2-z+1}{z^2+x^2+1}\ge 1}$$
37
votes
3answers
2k 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 ...
37
votes
1answer
777 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 $n \ge 3$: ...
36
votes
1answer
2k 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 ...
35
votes
15answers
3k views

Which one is bigger: $\;35{,}043 × 25{,}430\,$ or $\,35{,}430 × 25{,}043\;$?

Which of the two quantities is greater? Quantity A: $\;\;35{,}043 × 25{,}430$ Quantity B: $\;\;35{,}430 × 25{,}043$ What is the best and quickest way to get the answer without using ...
35
votes
10answers
4k views

Why does this way of solving inequalities work?

Here is what I had to prove. Question: For positive reals $a$ and $b$ prove that $a^2+b^2 \geq 2ab$. Here is how my teacher did it: First assume that it is in fact, true that $a^2+b^2 \geq ...
35
votes
2answers
1k views

Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator

Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator. I did in the following way. Are there other ways? Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then, ...
34
votes
3answers
1k views

A combinatorial proof of $n^n(n+2)^{n+1}>(n+1)^{2n+1}$?

The statement is 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 ...
33
votes
4answers
2k views

Which is bigger: $9^{9^{9^{9^{9^{9^{9^{9^{9^{9}}}}}}}}}$ or $9!!!!!!!!!$?

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) ...
32
votes
10answers
3k views

If both $a,b>0$, then $a^ab^b \ge a^bb^a$

Prove that $a^a \ b^b \ge a^b \ b^a$, if both $a$ and $b$ are positive.
32
votes
2answers
831 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 ...
32
votes
1answer
595 views

Proving the inequality $\frac{\log (1)}{1!}+\frac{\log ^2(2)}{2!}+\frac{\log^3(3)}{3!}+\cdots> \frac{\pi }{4}$

How to prove this inequality? $$\frac{\log (1)}{1!}+\frac{\log ^2(2)}{2!}+\frac{\log^3(3)}{3!}+\cdots> \frac{\pi }{4}$$ The left side looks vaguely like the series for $\exp(x)$: the terms ...
31
votes
6answers
962 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 ...
30
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 ...
30
votes
1answer
565 views

How prove this geometry inequality $R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2$

Zhautykov Olympiad 2015 problem 6 This links discusses the olympiad problem which none of students could solve , meaning it is very hard. Question: The area of a convex pentagon $ABCDE$ is $S$, ...
30
votes
2answers
1k views

How to prove this inequality? $\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}\ge 4+(x-y)^2$

Let $x,y,z>0$, and such $$4\le x+y+z\le 5.$$ Show that $$\dfrac{x^2}{y}+\dfrac{y^2}{z}+\dfrac{z^2}{x}\ge 4+(x-y)^2.$$ It seems that the condition $\dfrac{x^2}{y}+\dfrac{y^2}{z}+\dfrac{z^2}{x}\ge ...
29
votes
3answers
1k views

An inequality: $1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53$

$n$ is a positive integer, then $$1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53.$$ please don't refer to the famous $1+\frac1{2^2}+\frac1{3^2}+\dotsb=\frac{\pi^2}6$. I want to find a ...
29
votes
5answers
1k views

Which is bigger: $(\pi+1)^{\pi+1}$ or $\pi^{\pi+2}$?

I've been struggling for a while with the following problem: Which is bigger: $(\pi+1)^{\pi+1}$ or $\pi^{\pi+2}$? Needless to say software aid is not allowed. All manual calculations should be ...
29
votes
3answers
2k 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 ...
28
votes
6answers
2k 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 ...
27
votes
27answers
7k views

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

How can I prove that $xy\leq x^2+y^2$ for all $x,y\in\mathbb{R}$ ?
26
votes
6answers
2k 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 ...
26
votes
4answers
6k 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 ...
25
votes
4answers
1k views

Prove that $\sqrt{7}^{\sqrt{8}}>\sqrt{8}^{\sqrt{7}}$

show that $$\sqrt{7}^{\sqrt{8}}>\sqrt{8}^{\sqrt{7}}$$ and I found $$LHs-RHS=0.017\cdots$$ I have post this interesting problem Prove $(\dfrac{2}{5})^{\frac{2}{5}}<\ln{2}$ can someone suggest ...
25
votes
4answers
1k views

Prove that $2^{2^{\sqrt3}}>10$

With a computer or calculator, it is easy to show that $$ 2^{2^\sqrt{3}} = 10.000478 \ldots > 10. $$ How can we prove that $2^{2^{\sqrt3}}>10$ without a calculator?
25
votes
2answers
841 views

How to prove this inequality(7)?

let $x,y,z\in\mathbb{R}$, prove that $$4(x^6+y^6+z^6)+5(x^5y+y^5z+z^5x)\ge\dfrac{(x+y+z)^6}{27}$$ I do this sometimes, and I think this problem,is very hard,I hope someone can solve.Thank you By ...
24
votes
1answer
1k 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 ...
23
votes
4answers
734 views

Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$

Applying the Copson's inequality, I found: $$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where $\Psi^{(1)}(k)$ is the polygamma function. Is it known any ...
23
votes
6answers
2k views

Elegant Proof of a simple inequality

I'm looking for an elegant proof of the following identity: for $w_1,w_2,z_1,z_2\ge 0$, $w_1w_2+z_1z_2\le \max\{z_1,w_1\}\max\{z_2,w_2\}+\min\{z_1,w_1\}\min\{z_2,w_2\}$ The proof I currently have ...
23
votes
1answer
1k 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 > ...
23
votes
1answer
636 views

Is $\pi$ the best constant in this inequality?

Let $E$ be the set of completely monotonous functions on $[0,+\infty)$, that is $f \in C^\infty([0,+\infty))$ and $\forall\, n\geq 0,\forall\, x\geq 0,\quad(-1)^nf^{(n)}(x)\geq 0.$. For $f\in E$ and ...
22
votes
13answers
963 views

Which is greater, $98^{99} $ or $ 99^{98}$? [duplicate]

Which is greater, $98^{99} $ or $ 99^{98}$? What is the easiest method to do this which can be explained to someone in junior school i.e. without using log tables. I don't think there is an ...
22
votes
3answers
687 views

A (probably trivial) induction problem: $\sum_2^nk^{-2}\lt1$

So I'm a bit stuck on the following problem I'm attempting to solve. Essentially, I'm required to prove that $\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2} < 1$ for all $n$. I've been toiling ...