Questions tagged [a.m.-g.m.-inequality]
For questions about proving and manipulating the AM-GM inequality. To be used necessarily with the [inequality] tag.
1,314
questions
124
votes
25
answers
45k
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 ...
- 2,605
108
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 $$\frac1N\sum_k x_k ...
- 2,728
36
votes
3
answers
1k
views
Can we prove AM-GM Inequality using these integrals?
I came across these two results recently:
$$ \int_a^b \sqrt{\left(1-\dfrac{a}{x}\right)\left(\dfrac{b}{x}-1\right)} \: dx = \pi\left(\dfrac{a+b}{2} - \sqrt{ab}\right)$$
$$ \int_a^c \sqrt[3]{\left| \...
- 409
36
votes
6
answers
1k
views
New bound for Am-Gm of 2 variables
Today I'm interested by the following problem :
Let $x,y>0$ then we have :
$$x+y-\sqrt{xy}\leq\exp\Big(\frac{x\ln(x)+y\ln(y)}{x+y}\Big)$$
The equality case comes when $x=y$
My proof uses ...
user674646
25
votes
4
answers
2k
views
Inequality with five variables
Let $a$, $b$, $c$, $d$ and $e$ be positive numbers. Prove that:
$$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a}\geq\frac{a+b+c+d+e}{a+b+c+d+e-3\sqrt[5]{abcde}}$$
Easy to show ...
21
votes
4
answers
8k
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:
$\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$
$2(\sqrt{...
- 35.4k
20
votes
2
answers
1k
views
Is the AM-GM inequality the only obstruction for getting a specific sum and product?
This might be silly, but here it goes.
Let $P,S>0$ be positive real numbers that satisfy $\frac{S}{n} \ge \sqrt[n]{P}$.
Does there exist a sequence of positive real numbers $a_1,\dots,a_n$ such ...
- 24.4k
17
votes
6
answers
676
views
Prove the inequality $\sqrt\frac{a}{a+8} + \sqrt\frac{b}{b+8} +\sqrt\frac{c}{c+8} \geq 1$ with the constraint $abc=1$
If $a,b,c$ are positive reals such that $abc=1$, then prove that $$\sqrt\frac{a}{a+8} + \sqrt\frac{b}{b+8} +\sqrt\frac{c}{c+8} \geq 1$$ I tried substituting $x/y,y/z,z/x$, but it didn't help(I got the ...
user167045
17
votes
7
answers
7k
views
Prove $\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6$
If $a,b,c$ are non-negative numbers and $a+b+c=3$, prove that:
$$\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6.$$
Here's what I've tried:
Using Cauchy-Schawrz I proved that:
$$(3a + b^3)(...
- 35.4k
15
votes
4
answers
325
views
Finding maxima of a function $f(x) = \sqrt{x} - 2x^2$ without calculus
My question is how to prove that $f(x) = \sqrt x - 2x^2$ has its maximum at point $x_0 = \frac{1}{4}$
It is easy to do that by finding its derivative and setting it to be zero (this is how I got $x_0 ...
- 461
15
votes
2
answers
1k
views
Combined AM GM QM inequality
I came across this interesting inequality, and was looking for interesting proofs. $x,y,z \geq 0$
$$ 2\sqrt{\frac{x^{2}+y^{2}+z^{2}}{3}}+3\sqrt [3]{xyz}\leq 5\left(\frac{x+y+z}{3}\right) $$
Addendum....
- 4,858
14
votes
3
answers
1k
views
Prove $(a_1+b_1)^{1/n}\cdots(a_n+b_n)^{1/n}\ge \left(a_1\cdots a_n\right)^{1/n}+\left(b_1\cdots b_n\right)^{1/n}$
consider positive numbers $a_1,a_2,a_3,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$. does the following in-equality holds and if it does then how to prove it
$\left[(a_1+b_1)(a_2+b_2)\cdots(a_n+b_n)\right]^{...
- 671
14
votes
2
answers
562
views
The inequality $\,2+\sqrt{\frac p2}\leq\sum\limits_\text{cyc}\sqrt{\frac{a^2+pbc}{b^2+c^2}}\,$ where $0\leq p\leq 2$ is: Probably true! Provably true?
Let $p$ be a positive parameter in the range from $0$ to $2$.
Can one prove that
$$2 +\sqrt{\frac p2} \;\leqslant\;\sqrt{\frac{a^2 + pbc}{b^2+c^2}}
\,+\,\sqrt{\frac{b^2 +pca}{c^2+a^2}}\,+\,\sqrt{\...
- 5,362
13
votes
7
answers
440
views
How to prove the inequality $ \frac{a}{\sqrt{1+a}}+\frac{b}{\sqrt{1+b}}+\frac{c}{\sqrt{1+c}} \ge \frac{3\sqrt{2}}{2}$
How to prove the inequality $$ \frac{a}{\sqrt{1+a}}+\frac{b}{\sqrt{1+b}}+\frac{c}{\sqrt{1+c}} \ge \frac{3\sqrt{2}}{2}$$
for $a,b,c>0$ and $abc=1$?
I have tried prove $\frac{a}{\sqrt{1+a}}\ge \frac{...
- 225
13
votes
1
answer
328
views
Prove this Kenneth S. Williams inequality
If $0<a_1\le a_2\le \cdots\le a_n$, then the following inequality holds:
$$\frac{1}{2n^2a_n}{\sum_{1\le i < j\le n}^{} {(a_i-a_j)^2}}\le \frac{a_1+a_2+\cdots + a_n}{n}-\sqrt [n]{a_1 a_2 \cdots ...
- 91.6k
12
votes
2
answers
850
views
Solving Equation through inequalities.
If $x^6-12x^5+ax^4+bx^3+cx^2+dx+64=0$ has positive roots then find $a,b,c,d$.
I did something but that don't deserve to be added here, but what I thought before doing that is following:
For us, ...
- 3,291
12
votes
5
answers
581
views
proving :$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$.
Let $a,b,c>0$ how to prove that :
$$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$$
I find that
$$\ \frac{ab}{a^{2}+3b^{2}}=\frac{1}{\frac{a^{2}+3b^{2}}{ab}}=\frac{1}{\...
- 2,680
12
votes
1
answer
1k
views
Tighter inequality than Cauchy - Schwarz inequality
I just learned today that there can be a tighter result than AM-GM (Arithmetic Mean - Geometric Mean) inequality. In particular:
Let $a, b > 0$ then
\begin{equation}
\label{1}\tag{1}
\dfrac{a+b}{2}...
- 609
11
votes
5
answers
384
views
Prove that $a^4+b^4+1\ge a+b$.
Prove that
$$a^4+b^4+1\ge a+b$$
for all real numbers $a,b$.
What I've tried:
1.I checked how AM-GM may help but doesn't look like it's useful here.
I've tried:
$$(a^2+b^2)^2 -2(ab)^2+1 \ge a+b$$
...
user565804
11
votes
5
answers
1k
views
How to compare logarithms $\log_4 5$ and $\log_5 6$?
I need to compare $\log_4 5$ and $\log_5 6$.
I can estimate both numbers like $1.16$ and $1.11$. Then I took smallest fraction $\frac{8}{7}$ which is greater than $1.11$ and smaller than $1.16$ and ...
- 1,555
11
votes
5
answers
2k
views
British Maths Olympiad (BMO) 2002 Round 1 Question 3 Proof without Cauchy-Schwarz?
The question states:
Let $x,y,z$ be positive real numbers such that $x^2 + y^2 + z^2 = 1$
Prove that
$x^2yz + xy^2z + xyz^2 ≤ 1/3$
I have a proof of this relying on the fact that:
$x^2/3 +y^2/3 + ...
- 900
11
votes
3
answers
371
views
Minimum of $\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}$
What is the minimum of $$f(a,b,c):=\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{c+a}}+\sqrt{\dfrac{c}{a+b}}$$ where $a,b,c$ are positive real numbers?
When $a=b=c$, we have $f(a,b,c)=\dfrac{3}{\sqrt{2}}\...
- 2,587
11
votes
4
answers
531
views
Prove that if $a+b+c+d=4$, then $(a^2+3)(b^2+3)(c^2+3)(d^2+3)\geq256$
Given $a,b,c,d$ such that $a + b + c + d = 4$ show that $$(a^2 + 3)(b^2 + 3)(c^2 + 3)(d^2 + 3) \geq 256$$
What I have tried so far is using CBS:
$(a^2 + 3)(b^2 + 3) \geq (a\sqrt{3} + b\sqrt{3})^2 = ...
- 623
11
votes
1
answer
193
views
Tricky inequality involving 3 variables
Let $x, y$ and $z$ be three real numbers satisfying the following conditions:
$$0 < x \leq y \leq z$$
AND
$$xy + yz + zx = 3$$
Prove that the maximum value of $(x y^3 z^2)$ is $2.$
I tried ...
- 1,142
11
votes
2
answers
414
views
Prove this Generalizing AM-GM inequality
Let $n\ge 2$ and $a_{i} \ge 0,i=1,2,\cdots,n$, show that
$$(n-1)^{n-1}(a^n_{1}+a^n_{2}+\cdots+a^n_{n})+n^na_{1}a_{2}\cdots a_{n}\ge (a_{1}+a_{2}+\cdots+a_{n})^n$$
When $n=2$,
$$a^2_{2}+a^2_{2}+4a_{...
- 91.6k
10
votes
2
answers
287
views
Typical Olympiad Inequality? If $\sum_i^na_i=n$ with $a_i>0$, then $\sum_{i=1}^n\left(\frac{a_i^3+1}{a_i^2+1}\right)^4\geq n$
Let $\sum_i^na_i=n$, $a_i>0$. Then prove that $$ \sum_{i=1}^n\left(\frac{a_i^3+1}{a_i^2+1}\right)^4\geq n $$
I have tried AM-GM, Cauchy-Schwarz, Rearrangement etc. but nothing seems to work. The ...
- 397
10
votes
2
answers
1k
views
Min $P=\frac{x^5y}{x^2+1} +\frac{y^5z}{y^2+1} +\frac{z^5x}{z^2+1}$
Given $x,y,z$ are positive numbers such that $$x^2y+y^2z+z^2x=3$$
Find the minium of value: $$P=\frac{x^5y}{x^2+1} +\frac{y^5z}{y^2+1} +\frac{z^5x}{z^2+1} $$
My Attempt:
$$P= x^3y+y^3z+z^3x - \left( ...
- 414
10
votes
2
answers
705
views
Proving the inequality $4\ge a^2b+b^2c+c^2a+abc$
So, a,b,c are non-negative real numbers for which holds that $a+b+c=3$.
Prove the following inequality: $$4\ge a^2b+b^2c+c^2a+abc$$
For now I have only tried to write the inequality as $$4\left(\frac{...
- 3,330
10
votes
1
answer
689
views
SOS: Proof of the AM-GM inequality
"A basic strategy in tackling inequalities of few variables is to write things into sum of squares..." is a quote from an answer, intended as commenting the post entitled "Can this inequality proof be ...
- 5,362
10
votes
0
answers
168
views
Solving the system $\tan x+\sin y+\sin z=3x$, $\sin x+\tan y+\sin z=3y$, $\sin x+\sin y+\tan z=3z$
If $x,y,z\in (0,\frac{\pi}2 )$, find all solutions to:
$$\begin{cases} \tan x+\sin y+\sin z=3x \\
\sin x+\tan y+\sin z=3y \\
\sin x+\sin y+\tan z=3z\end{cases}$$
This question was deleted here: https:/...
9
votes
1
answer
517
views
Proof of $(1-x)x^n \leq \frac{n^n}{\left(n+1\right)^{n+1}}$ without use of derivatives
If $x \in \left[0,1\right]$ and $n \in \mathbb{Z}^+$, is it possible to show $$(1-x)x^n \leq \frac{n^n}{\left(n+1\right)^{n+1}}$$ without use of derivatives?
With derivative it's smooth:
Let $f(x)=(...
- 1,622
9
votes
3
answers
305
views
For $a\geq2$, $b\geq2$ and $c\geq2$, prove that $\left(a^3+b\right)\left(b^3+c\right)\left(c^3+a\right)\geq125 abc$
For $a\geq2$, $b\geq2$ and $c\geq2$, prove that
$$(a^3+b)(b^3+c)(c^3+a)\geq 125 abc.$$
My try:
First I wrote the inequality as
$$\left(a^2+\frac{b}{a}\right) \left(b^2+\frac{c}{b}\right) \left(...
user596786
9
votes
1
answer
389
views
USA $2011$ contest inequality problem, proving $\frac{ab+1}{(a+b)^2}+\frac{bc+1}{(b+c)^2}+\frac{ca+1}{(c+a)^2}\ge 3$, under given condition.
If $a^2+b^2+c^2+(a+b+c)^2\le4$, then $$\frac{ab+1}{(a+b)^2}+\frac{bc+1}{(b+c)^2}+\frac{ca+1}{(c+a)^2}\ge 3.$$
My attempt:
From the given criteria, one can easily obtain that $$(a+b)^2+(b+c)^2+(c+a)^...
user249332
9
votes
1
answer
5k
views
Proving the AM-GM Inequality with Lagrange Multipliers
Exercise: Let $x_1,x_2,...,x_n$ be real positive numbers. Prove the arithmetic-geometric mean inequality, $(x_1x_2...x_n)^{1/n}\le (x_1+x_2+...+x_n)/n$.
Hint: Consider the function $f(x_1,x_2,...,x_n)...
- 1,543
9
votes
3
answers
3k
views
True or false: $a^2+b^2+c^2 +2abc+1\geq 2(ab+bc+ca)$
Is this inequality true?
$a^2+b^2+c^2 +2abc+1\ge2(ab+bc+ca)$, where $a,b,c\gt0$.
Can you find a counterexample for this or not?
user85046
9
votes
1
answer
235
views
Intermediate inequalities; is there a way to know if you're getting a "bad deal"?
Let's say I want to prove a contest-style inequality $f(a, b, c) + g(a, b, c) \ge h(a, b, c)$ in some $S \subseteq \mathbb{R}^3$. Suppose I want to make the RHS simpler by applying the AM-GM ...
- 22.7k
8
votes
5
answers
3k
views
Prove that $xy \leq\frac{x^p}{p} + \frac{y^q}{q}$
OK guys I have this problem:
For $x,y,p,q>0$ and $ \frac {1} {p} + \frac {1}{q}=1 $ prove that $ xy \leq\frac{x^p}{p} + \frac{y^q}{q}$
It says I should use Jensen's inequality, but I can't figure ...
- 982
8
votes
2
answers
1k
views
Largest possible value of trigonometric functions
Find the largest possible value of
$$\sin(a_1)\cos(a_2) + \sin(a_2)\cos(a_3) + \cdots + \sin(a_{2014})\cos(a_1)$$
Since the range of the $\sin$ and $\cos$ function is between $1$ and $-1$, shouldn'...
- 2,466
8
votes
6
answers
344
views
Prove $a^2 + b^2 + c^2 + ab + bc +ca \ge 6$ given $a+b+c = 3$ for $a,b,c$ non-negative real.
I want to solve this problem using only the AM-GM inequality. Can someone give me the softest possible hint? Thanks.
Useless fact: from equality we can conclude $abc \le 1$.
Attempt 1:
Adding $(ab + ...
- 2,261
8
votes
4
answers
483
views
Proving :$\frac{1}{2ab^2+1}+\frac{1}{2bc^2+1}+\frac{1}{2ca^2+1}\ge1$
Let $a,b,c>0$ be real numbers such that $a+b+c=3$,how to prove that? :
$$\frac{1}{2ab^2+1}+\frac{1}{2bc^2+1}+\frac{1}{2ca^2+1}\ge1$$
- 2,680
8
votes
4
answers
294
views
Theorem about two real numbers
My question is:
$a,b$ are two positive real numbers such that their product is constant,equal to $k$ say. Prove: the sum $a+b$ is minimum if and only if $a = b= \sqrt k$.
Can this be solved using $A....
- 955
8
votes
5
answers
632
views
Given positive numbers $x_1,...,x_n$ such that $x_1\cdots x_n=1$ prove that $\frac{1}{n-1+x_1}+\cdots+\frac{1}{n-1+x_n}\le 1$
Given positive numbers $x_1,\dots,x_n$ such that $x_1\cdots x_n=1$ prove that $\frac{1}{n-1+x_1}+\cdots+\frac{1}{n-1+x_n}\le 1$.
I tried solving this question through substitution, e.g. $a_1=\sqrt[n]{...
- 1,167
8
votes
3
answers
1k
views
Challenging inequality: $abcde=1$, show that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{33}{2(a+b+c+d+e)}\ge{\frac{{83}}{10}}$
Let $a,b,c,d,e$ be positive real numbers which satisfy $abcde=1$. How can one prove that:
$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} +\frac{1}{e}+ \frac{33}{2(a + b + c + d+e)} \ge{\frac{{...
- 4,828
8
votes
3
answers
270
views
Prove the inequality $\sum_{cyc}\frac{a^2}{\sqrt{(b+c)(b^3+c^3)}}\geq\frac 32$ where $a,b,c$ are positive reals.
Let $a,b,c$ be positive real numbers, prove that $$\sum_{cyc}\frac{a^2}{\sqrt{(b+c)(b^3+c^3)}}\geq\frac 32.$$
The problem is from an inequality handout. Here is my attempt to solve the problem:
I ...
- 3,896
8
votes
2
answers
290
views
If $a+b+c+d=16$, then $(a+\frac{1}{c})^2+(c+\frac{1}{a})^2 + (b+\frac{1}{d})^2 + (d+\frac{1}{b})^2 \geq \frac{289}{4}$
If $a,b,c,d$ are positive integers and $a+b+c+d=16$, prove that
$$\left(a+\frac{1}{c}\right)^2+\left(c+\frac{1}{a}\right)^2+\left(b+\frac{1}{d}\right)^2+\left(d+\frac{1}{b}\right)^2 \geq \frac{289}{...
- 519
8
votes
3
answers
151
views
Prove that $\frac{ab}{a+b} + \frac{cd}{c+d} \leq \frac{(a+c)(b+d)}{a+b+c+d}$
$$\frac{ab}{a+b} + \frac{cd}{c+d} \leq \frac{(a+c)(b+d)}{a+b+c+d}$$
I tried applying a.m. g.m inequality to l.h.s and tried to find upper bound for l.h.s and lower bound for r.h.s but i am not getting ...
- 129
8
votes
1
answer
166
views
Inequality, how to know intuition behind it
I was solving the following inequality
For $a$, $b$, $c$ and $d$ being positive real numbers
which goes as
$$
\frac{a}{a+b} + \frac{b}{b+c} + \frac{c}{c+d} + \frac{d}{d+a} \leq \frac{a}{b+c} + \frac{...
- 770
8
votes
1
answer
652
views
Proving with AM-GM Inequality [duplicate]
$$\frac{4}{abcd}\geq\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}$$
Given: $a+b+c+d=4$ and $a$, $b$, $c$ abd $d$ are positives.
How to prove the above inequality using Arithmetic Geometric Mean ...
- 1,016
7
votes
4
answers
2k
views
if : $abc=8 $ then : $(a+1)(b+1)(c+1)≥27$
if : $$abc=8 :a,b,c\in \mathbb{R}_{> 0}$$
then :
$$(a+1)(b+1)(c+1)\ge 27.$$
My try :
$$(a+1)(b+1)(c+1)=1+(a+b+c)+(ab+ac+bc)+abc$$
$$(a+1)(b+1)(c+1)=1+(a+b+c)+(ab+ac+bc)+8, $$
then?
- 4,752
7
votes
6
answers
770
views
Prove that $\frac{\ 1}{a^2+b^2+ab}+\frac{\ 1}{b^2+c^2+bc}+\ \frac{\ 1}{c^2+a^2+ca}\ge\ \ \frac{\ 9}{\left(a+b+c\right)^2}$.
Let $a,b,c$ be positive real numbers. Prove that $$\frac{\ 1}{a^2+b^2+ab}+\frac{\ 1}{b^2+c^2+bc}+\ \frac{\ 1}{c^2+a^2+ca}\ge\ \ \frac{\ 9}{\left(a+b+c\right)^2}.$$
I want to prove the inequality with ...
- 3,896