Questions on proving and manipulating inequalities.

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87
votes
1answer
3k 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: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ ...
65
votes
3answers
2k 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 ...
53
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 ...
45
votes
2answers
2k 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.
44
votes
2answers
909 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) ...
40
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 ...
40
votes
2answers
873 views

This is stupid but I have a bad cold with cough

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 permutaion? It does not work well if any ...
38
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 ...
37
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 ...
36
votes
3answers
1k 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 $$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$$ ...
34
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 ...
33
votes
18answers
3k 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 ...
33
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
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 ...
31
votes
6answers
947 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 ...
31
votes
2answers
565 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 ...
30
votes
7answers
3k 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 ?
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
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 ...
29
votes
1answer
636 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 $n \ge 3$: ...
29
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 ...
27
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 ...
27
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 ...
26
votes
3answers
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) ...
26
votes
9answers
975 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 ...
25
votes
27answers
5k 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 $x,y\in\mathbb{R}$ ?
25
votes
6answers
1k 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 ...
23
votes
3answers
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 ...
23
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?
23
votes
1answer
830 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
2answers
768 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 ...
22
votes
7answers
655 views

How prove this $f(n)\le f(n+1)$ where $f(n)=\sum_{k=1}^{n}\frac{n}{n^2+k^2}$

let $$f(n)=\sum_{k=1}^{n}\dfrac{n}{n^2+k^2}$$ prove or disprove $$f(n)\le f(n+1)$$ this inequality is found when I deal this follow limit: ...
22
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 ...
22
votes
1answer
587 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
2answers
1k views

An information theory inequality which relates to Shannon Entropy

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 ...
21
votes
4answers
4k 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 ...
21
votes
2answers
483 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 ...
21
votes
2answers
439 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
8answers
865 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 ...
20
votes
4answers
558 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 know any ...
20
votes
2answers
911 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 ...
20
votes
2answers
302 views

How prove this inequality $2a^ab^bc^cd^d\ge ac+bd$

Let $a,b,c,d$ be positive numbers such that $a+b+c+d=2$. Show that $$2a^ab^bc^cd^d\ge ac+bd$$ My try: I think maybe I can use this inequality $$(1+x)^n\ge 1+nx \hspace{12pt} (n>1)$$ then I can't ...
20
votes
2answers
336 views

How prove that:$\varphi(2)+\varphi(3)+\varphi(4)+\cdots+\varphi(n)\ge\frac{n(n-1)}{4}+1$

Prove that for $n\ge 3$, $$\varphi(2)+\varphi(3)+\varphi(4)+\cdots+\varphi(n)\ge\dfrac{n(n-1)}{4}+1$$ where $\varphi$ is the Euler's totient function I think we must use this ...
20
votes
0answers
841 views
+50

Prove this inequality with $xyz\le 1$

if $x,y,z>0$ and $\color{red}{xyz\le 1}$, show that $$\color{blue}{\dfrac{x^2-x+1}{x^2+y^2+1}+\dfrac{y^2-y+1}{y^2+z^2+1} +\dfrac{z^2-z+1}{z^2+x^2+1}\ge 1}$$ Sorry,This is not 2015 TST ...
20
votes
2answers
631 views

Proving that the triangle inequality holds for a metric on $\mathbb{C}$

Show that $(X,d)$ is a metric space where $X =\Bbb C $ and the distance function is defined as: $$d(x,y) = \frac {2|x-y|}{\sqrt {1+|x|^2} + \sqrt {1 + |y|^2}}, \text{ for } x,y \in \Bbb C.$$ I ...
19
votes
4answers
310 views

How to prove this inequality $\frac{a_{1}a_{2}+a_{2}a_{3}+\cdots+a_{n-1}a_{n}}{a^2_{1}+a^2_{2}+\cdots+a^2_{n}}\le\cos{\frac{\pi}{n+1}}$

Let $a_{1},a_{2},\cdots,a_{n},n\ge 2$ be real numbers,show that $$\dfrac{a_{1}a_{2}+a_{2}a_{3}+\cdots+a_{n-1}a_{n}}{a^2_{1}+a^2_{2}+\cdots+a^2_{n}}\le\cos{\dfrac{\pi}{n+1}}$$ I think this ...
19
votes
4answers
1k views

Prove $\sqrt{x^2+yz+2}+\sqrt{y^2+zx+2}+\sqrt{z^2+xy+2}\ge 6$, given $x+y+z=3$ and $x,y,z\ge0$

Let $x+y+z=3,x,y,z\ge 0$,show that $$\sqrt{x^2+yz+2}+\sqrt{y^2+zx+2}+\sqrt{z^2+xy+2}\ge 6$$ Additional information I have seen the following problem: $x,y,z>0,x+y+z=3$, prove that ...
19
votes
1answer
395 views

Existence of two real numbers satisfying $f(x-f(y))>yf(x)+x$

Let $f:\mathbb{R} \longrightarrow \mathbb{R}$ be a function. Is it always the case that for some $x,y \in \mathbb R$, the inequality $f(x-f(y))>yf(x)+x$ holds? Thanks in advance.
19
votes
2answers
500 views

Maximizing curious symmetric function from simple combinatorics

A curious symmetric function crossed my way in some quantum mechanics calculations, and I'm interested its maximum value (for which I do have a conjecture). (This question has been posted at ...
18
votes
3answers
925 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 ...