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.
75
questions with no upvoted or accepted answers
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:/...
5
votes
0
answers
237
views
Existence of a factorisation proof of the AM-GM inequality
Consider the following formulation of the AM-GM Inequalities:
$$\large f_n(a_1, a_2, a_3,\cdots,a_n) := \left(\sum_{k=1}^n a_k^n\right) - n \prod_{k=1}^n a_k \geqslant 0$$
As required by the classical ...
- 4,203
4
votes
1
answer
204
views
Prove or disprove that the inequality is valid if $x,y,z,u$ are positive numbers and $x+y+z+u=2$.
Prove or disprove that the inequality $$ \dfrac{x^2}{\left(x^2+1\right)^2}+\dfrac{y^2}{\left(y^2+1\right)^2}+\dfrac{z^2}{\left(z^2+1\right)^2}+\dfrac{u^2}{\left(u^2+1\right)^2} \leq \dfrac{16}{25}$$ ...
- 297
4
votes
0
answers
258
views
An inequality using column sums of inverse matrices
I want to prove a matrix analogue to inequality $\left(\frac{1-x(1-\alpha)}{\alpha}\right)^{\alpha} x^{1-\alpha}$ for $\alpha \in [0,1)$ and $x \in [0,1]$, which has a nice proof using GM-AM, as shown ...
- 79
4
votes
0
answers
1k
views
Proof by induction of AM-GM inequality
I recently came up with
a proof
by simple induction
of the
arithmetic mean -
geometric mean inequality
that I haven't found here.
I'm sure it isn't new.
My questions:
(1) Is this correct?
(2) Is this ...
- 104k
4
votes
0
answers
200
views
What is the content of Young's inequality?
On the one hand, Young's inequality, in the form of
$$
ab \leq \frac{a^p}{p}+\frac{b^q}{q}
$$
where $p$ and $q$ are Hölder conjugates, can be seen to be easily rearranged to be a restatement of ...
- 15.9k
3
votes
0
answers
102
views
Maximum value of a variable in system of two equations
Consider the system of equations
$$ 3a + 2b + c+ d =14 $$
$$ a^{2} + b^{2} + c^{2} + d^{2} =14.$$
Is there any way to find maximum value of $d$ using AM-GM inequality. I am not even able to think, how ...
- 1,243
3
votes
0
answers
104
views
min of $c$ in such that $\frac{\sum^n_{i=1}a_i}{n\sqrt[n]{\prod^n_{i=1}a_i}}\leq \left(\frac{\sqrt{a/b}+\sqrt{b/a}}2\right)^c$ for all $a_i\in [a,b]$.
Positive integer $n\geq 2$ is given. All $a_i\in[a,b]$($0<a<b$). What's the minimum value of $c$ such that $$\tag{1}\frac{\sum^n_{i=1}a_i}{n\sqrt[n]{\prod^n_{i=1}a_i}}\leq \left(\frac{\sqrt{\...
- 31
3
votes
0
answers
95
views
How to prove $\dfrac{a+b}{2}\geq\sqrt{ab}$ using ellipse
Noted that ellipse properties $d_1+d_2=2D$, focal length, $f=c$ and radius of minor axis $=r$.
Let $d_1=a;d_2=b$
If $ab$ is not maximum, then $\sqrt{ab}$ not maximum
W.l.o.g, prove max $ab=D^2$
$ab=\...
- 135
3
votes
0
answers
246
views
Partial Proof of an inequality by Vasile Cirtoaje
Claim
We want to show [1]:
Let $0.36\leq x\leq 0.5$ and $1\leq k\leq n$ two naturals numbers with $n\geq 10^{10}$ then we have :
$$P(k)=(1-x)^{(2x)^{1+\frac{k}{n}}}+x^{(2(1-x))^{1+\frac{k}{n}}}\leq 1\...
- 3,333
3
votes
0
answers
141
views
On Proving $ a^6 + b^6 + c^ 6 − 3a ^2 b^ 2 c^ 2 + 2(a^ 2 + bc)(b ^2 + ca)(c^ 2 + ab) ≥ 0 $ where $a,b,c$ are real numbers.
On this problem, I made some observations and then started to solve.
Here one important note is that the inequality is trivial for $a,b,c\ge0$ and $a,b,c\le0$ and the inequality is symmetric.
Another ...
- 1,506
3
votes
2
answers
152
views
A Hard Inequality
Given that $x,y,z$ are positive real numbers such that $2x+4y+7z=2xyz$, find the minimum of $L=x+y+z$.
Does anybody have a solution that is purely algebraic?
I was only able to solve it with Lagrange ...
user485639
2
votes
0
answers
68
views
Prove or disprove that the inequality is true if x, y, z are positives and $xyz=8^3$.
Prove or disprove that the inequality $$ \dfrac{1}{\sqrt{x+1}}+\dfrac{1}{\sqrt{y+1}}+\dfrac{1}{\sqrt{z+1}} \geq 1$$ is true if x, y, z are positives and $$xyz=8^3.$$
First, let us rewrite the ...
- 297
2
votes
0
answers
59
views
Finding a limit using AM-GM?
Doing some calculus papers before going university and I found this question:
Find $\lim_{x\to\infty} \left[\frac{1}{3} \left(3^\frac{1}{x} + 8^\frac{1}{x} + 9^\frac{1}{x} \right)\right]^x$
My ...
- 93
2
votes
0
answers
92
views
Find bounds of $\prod_{i=1}^k (x_i-x_{i+1})$ where $\sum_{i=1}^k x^2_i=1$
Let $x_1,x_2,...,x_k$ be real numbers such that $\sum_{i=1}^k x^2_i=1$. Determine the minimum and maximum (if there is) value of $$\prod_{i=1}^k (x_i-x_{i+1})$$ and determine all values of $(x_1,x_2,...
- 129
2
votes
1
answer
94
views
Prove that for positive reals $x,y,z$, $x^6+y^6+z^6 + 6x^2y^2z^2 \geq 3xyz(x^3+y^3+z^3)$.
I am not sure if the inequality is true. My first attempt was to try AM-GM inequality in clever ways. I also tried Schur's inequality which gives
$$
x^6+y^6+z^6 + 6x^2y^2z^2 \geq (x^2+y^2+z^2)(x^2y^2+...
- 29
2
votes
0
answers
78
views
Prove that for all $a,b,c \in \mathbb{R}^+$, $\sqrt[3]{\frac ab} + \sqrt[5]{\frac bc} + \sqrt[7]{\frac ca} > \frac 52$
Prove that for all $a,b,c \in \mathbb{R}^+$, $\sqrt[3]{\dfrac ab} + \sqrt[5]{\dfrac bc} + \sqrt[7]{\dfrac ca} > \dfrac 52$.
My thought process along with the proof: Since we're dealing with ...
- 839
2
votes
0
answers
89
views
$\sum\sqrt{\frac{2a}{b+c}}\le\sqrt[3]{9\sum\frac{a}{b}}$
Let $a$, $b$ and $c$ be positive numbers. Prove that:
$$\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}}\le\sqrt[3]{9\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$$
It is from ...
user875449
2
votes
3
answers
159
views
Inequality of arithmetic mean of two sets
If $a,b>0$ and $Q=\{x_1, x_2, x_3,..., x_a\}$ a subset of the natural numbers $1, 2, 3,..., b$ such that, for $x_i+x_j<b+1$ with $1 ≤ i ≤ j ≤ a$, then $x_i+x_j$ is also an element of Q. Prove ...
- 93
2
votes
0
answers
69
views
Refinement inequality of : $\sqrt{x}+x^{\frac{x}{x+1}}\geq x+1$
Related to New bound for Am-Gm of 2 variables we have :
Let $x\geq 5$ be a real number then we have :
$$\sqrt{x}+x^{\frac{x}{x+1}}\geq \frac{x^2+1}{x+1}\Bigg(\frac{x^{\frac{x}{x+1}}}{x^{\...
- 3,333
2
votes
0
answers
77
views
Proving that $\sqrt{k^2 a^2+bc}+\sqrt{k^2 b ^2+ac}+\sqrt{k^2 c ^2+ab}\le(k+\frac12)\cdot(a+b+c)$
Let $k\ge 1$. I want to prove that for all $a,b,c\geq 0$ we have $$\sqrt{k^2 a^2+bc}+\sqrt{k^2 b ^2+ac}+\sqrt{k^2 c ^2+ab}\le\left(k+\frac12\right)\cdot(a+b+c).$$
My attempt: I tried to use that $\...
2
votes
1
answer
144
views
optimization with strict inequality of variables
Maximize $f(x_1,x_2, x_3) = x_{2}+x_{3} - (x_{2}^2+x_{3}^2)$ given $\sum_{i=1}^{3}x_{i} = 1$ and $x_{i}>0$ for $i=1,2,3$. I f I assume that $x_{i}\geq0$ for $i=1,2,3$ then the solution is $x_2 = ...
- 367
2
votes
1
answer
98
views
Cauchy-Schwarz sanity check
Cauchy Schwarz say that $$\mid x_1y_1\mid +...+ \mid x_ny_n\mid \leq \sqrt{x_1^2+...+x_n^2}\sqrt{y_1^2+...+y_n^2}.$$ (This follows from the popular proof using AM-GM.)
This is Holder's inequality ...
- 1,170
2
votes
1
answer
191
views
Find $max(\prod{a_i})$ given that $\sum a_i=2017$ for $n$ number of positive integers from $a_1, a_2, \cdots, a_n$
Find the maximum value of $(\prod{a_i})$ given that $\sum a_i=2017$ for $n$ number of positive integers from $a_1, a_2, \cdots, a_n$
I don't understand how to do it.
I had thought of proceeding by ...
- 3,304
2
votes
0
answers
91
views
Weaker version of the $A.M.\;-\;G.M.$ inequality
It is a well known fact that, for $\mathbf{x} \in R^k$ with $\mathbf{x}=(x_1,...,x_k) \geq 0$, $A(\mathbf{x})=G(\mathbf{x})$ iff $x_1=x_2=...=x_n$, where $A(\mathbf{x})$ and $G(\mathbf{x})$ are the ...
- 307
1
vote
1
answer
73
views
Inequality proof with AM-GM
I'm new to this community. Do you think my solution is correct?
The problem is:
$$a^2+b^2+c^2+ab+bc+ca\geq6, a+b+c=3$$
My solution:
$$a^2+b^2+c^2+2ab+2bc+2ca\geq3(ab+bc+ca)\rightarrow$$
$$3\geq ab+bc+...
- 21
1
vote
0
answers
113
views
Find the maximum value of $ab+bc+cd+de+ef+fa,$ given that $a+b+c+d+e+f=1$
After looking at this post, I framed the following question.
If $a,b,c,d,e,f$ are positive real numbers such that $a+b+c+d+e+f=1$, then find the minimum and maximum value of $ab+bc+cd+de+ef+fa$
My ...
- 7,040
1
vote
0
answers
26
views
Sequence of geometric mean subtracted by arithmetic mean
Let $a_1,a_2,a_3,\dots$ be a sequence of positive numbers. Define $$G_n=\sqrt[n]{a_1a_2\dots a_n}~\text{and}~A_n=\frac{a_1+\dots+a_n}{n}.$$ We are supposed to use the result $$u^av^b\leq au+bv \tag{$*$...
- 311
1
vote
0
answers
52
views
Inequality related to AM-GM?
Let $a$ and $b$ be two positive numbers such that $a+b=1$. I am supposed to show that $u^av^b\leq au+bv$ for all positive $u$ and $v$.
It is known that $\ln x \leq x$ for all positive $x$, so I ...
- 311
1
vote
0
answers
140
views
prove that inequality holds for all reals if and only if $|k| \leq 2n$
Prove that $$x_1^2+x_2^2+\ldots+x_{2n}^2+k \cdot x_1 \cdot x_2 \cdot \ldots \cdot x_{2n}\geq 0$$
for all reals $x_1, x_2, \ldots, x_{2n}$ if and only if $|k| \leq 2n$, where $n$ and $k$ integers, $n &...
1
vote
1
answer
74
views
An Application of Cauchy Schwarz - AM-GM in Discrete Probability Measures
Suppose $p_m \geq 0$ and $\sum_{m \in \mathbf{Z}} p_m =1 .$ That is $p$ is a probability measure on integers. Then how can I show (is it true) that
$$ \sum_{m \in \mathbf{Z}} (p_m + p_{m+1} + p_{m+2})^...
- 158
1
vote
0
answers
40
views
How can i find minimum value of this inequality
$a_1,a_2,...,a_n$ are 8 distinct positive integers. $b_1,b_2,...,b_n$ are another 8 distinct positive integers ($a_i,b_j$ are not necessarily y distinct for $i, j = 1, 2, ...8$).Enter
the smallest ...
1
vote
1
answer
76
views
Prove $\sum_{j=1}^{n^2} \log_n(2j-1)\leq2n^2 $
Deduce whether the statement is true or false.
Suppose $n\in \mathbb N \setminus \{0,1\}$. Then, $$\sum_{j=1}^{n^2} \log_n(2j-1)\leq2n^2 $$
I would like to ask what inequality I can apply or any ...
- 133
1
vote
0
answers
76
views
Maximizing $\dfrac{4 \sqrt{a} + 6 \sqrt{b} + 12 \sqrt{c}}{\sqrt{abc}}$ over $a+b+c=4abc$ and $a,b,c>0$
Let $a,$ $b,$ $c$ be positive real numbers such that $a + b + c = 4abc.$ Find the maximum value of
$$\dfrac{4 \sqrt{a} + 6 \sqrt{b} + 12 \sqrt{c}}{\sqrt{abc}}.$$
First, I tried using $\dfrac{4 \sqrt{...
- 748
1
vote
1
answer
71
views
Hard AM-GM Inequality
Find the sum of all positive integers $n,$ where the inequality
$\sqrt{a + \sqrt{b + \sqrt{c}}} \ge \sqrt[n]{abc}$ holds for all nonnegative real numbers $a,$ $b,$ and $c.$
I tried squaring both sides ...
- 77
1
vote
1
answer
53
views
When does this inequality hold? $\sum_{cyc}\frac{a^2+(a+b+d)c}{a^3+3bcd}\ge\frac{(a+b+c+d)^2}{a^3+b^3+c^3+d^3}$
For positive real numbers $a,b,c,d$, prove the inequality $$\frac{a^2+(a+b+d)c}{a^3+3bcd}+\frac{b^2+(a+b+c)d}{b^3+3acd}+\frac{c^2+(b+c+d)a}{c^3+3abd}+\frac{d^2+(a+c+d)b}{d^3+3abc}\ge\frac{(a+b+c+d)^2}{...
- 417
1
vote
0
answers
139
views
Is there a generic reversing of the Arithmetic Mean – Geometric Mean inequality？
Suppose $x_1,\dots,x_n$ are all positive real numbers. The Arithmetic Mean is $\frac{\sum_{i=1}^n x_i}{n}$. The Geometric Mean is $\sqrt[n]{\prod_{i=1}^n x_i}$. Is there a constant $C$ depending on $n$...
- 109
1
vote
0
answers
56
views
Chrystal's proof of the arithmetic-mean-geometric-mean inequality
I was looking through
Chrystal's "Algebra, Part II"
(1900),
available for free on the web,
and I noticed that
it had a proof of the
arithmetic-mean-geometric-mean
inequality
by induction.
I ...
- 104k
1
vote
0
answers
209
views
Weighted quadratic mean inequality
It is very well-known that given positive real numbers $a,b$, then
$$
\dfrac{a+b}{2} \le \sqrt{\dfrac{a^2+b^2}{2}} ,
$$
and equality holds if and only if $a=b$.
Is there an analogous "weighted&...
- 7,535
1
vote
0
answers
49
views
how to prove this inequality which gives upper bound of inner product $|\langle A_{t}(x),x\rangle|$
how to obtain this inequality
$|\langle A_{t}(x),x\rangle| \leq C( f_{t}+f_{t}^\frac{p_{0}}{2}+(1+|x|_{H})^{p_{0}}+|x|^{\alpha}_{V}+|x|^{\alpha}_{V}+|x|^{\alpha}_{V}|x|^{\beta}_{H})$.
I am reading a ...
- 41
1
vote
0
answers
31
views
Convergence/divergence of sum of finite products of functions of reciprocals of natural numbers
Convergence/divergence of sum of finite products of functions of reciprocals of natural numbers where the product of the functions equals the identity function.
Yes that was quite the mouthful.
...
- 15.8k
1
vote
1
answer
116
views
Variation of Nesbitt Inequality with the geometric mean
The problem is the following:
Prove the inequality: $ \frac{\sqrt{pq}}{p + q + 2r}+\frac{\sqrt{qr}}{q + r+2p}+\frac{\sqrt{pr}}{p + r+2q}≤3/4 $ for $p, q, r>0$ real numbers.
I could prove a weaker ...
- 93
1
vote
0
answers
38
views
Find $W:= \frac{xy}{z}+\frac{yz}{x}+\frac{xz}{y} \rightarrow \inf$
I've got two similar problems:
Find
\begin{align}
&W:= \frac{xy}{z}+\frac{yz}{x}+\frac{xz}{y} \rightarrow \inf \\
&x + y + z = 1 \\
&x, y, z > 0
\end{align}
and
\begin{align}
&W:...
- 470
1
vote
0
answers
67
views
Inequality $3\cos(\frac{1}{3}\tan(\frac{1}{3}\sin(\frac{1}{3})))\leq\sum_{cyc}\cos(a\tan(b\sin(c)))\leq 3$
I'm proud to present one inequality of my work :
Let $a,b,c>0$ such that $a+b+c=1$ then we have :
$$3\cos(\frac{1}{3}\tan(\frac{1}{3}\sin(\frac{1}{3})))\leq\sum_{cyc}\cos(a\tan(b\sin(c)))\leq ...
- 3,333
1
vote
1
answer
50
views
Tough substitution inequality
Prove that if $x, y, z >0$ and $xyz=x+y+z+2$, then
$$
\sqrt{x}+\sqrt{y}+\sqrt{z} \leq \frac{3}{2}\sqrt{xyz}.
$$
By the way, the first equation implies the existence of positive $a, b, c$ such ...
- 419
0
votes
0
answers
23
views
Three-Variable Inequality
Prove that if $a$, $b$, and $c$ are positive real numbers, then
$$\sqrt{a^2 + ab + b^2} + \sqrt{a^2 + ac + c^2} + \sqrt{b^2 + bc + c^2} \ge \sqrt{3} (\sqrt{ab} + \sqrt{ac} + \sqrt{bc}).$$
When does ...
- 87
0
votes
1
answer
37
views
$\left\{(x_1,x_2)\mid \left|f(x_1,x_2)-f(0,0)\right|<\frac{1}{2}\right\}=\left\{(0,0)\right\}\cup\left\{(x_1,x_2)\mid x_1\neq\pm x_2\right\}$
I am reading "Introduction to Set Theory and Topology" (in Japanese) by Kazuo Matsuzaka.
Problem 18 on p.194
Let $f$ be a function from $\mathbb{R}\times\mathbb{R}$ to $\mathbb{R}$ such ...
- 7,456
0
votes
1
answer
74
views
Prove that the inequality is valid if $x,y,z$ are positive numbers and $xyz=1.$
Is given that $x,y,z$ are positive numbers and $xyz=1$, prove that
$$\dfrac{\dfrac{1}{x}}{\sqrt{z^2+1}}+\dfrac{\dfrac{1}{y}}{\sqrt{x^2+1}}+\dfrac{\dfrac{1}{z}} {\sqrt{y^2+1}}>\sqrt{2}.$$
What have ...
- 297
0
votes
0
answers
32
views
Please help me on how to prove the equality case of the inequality
goal:
for n=2,3,4....
prove that given x1x2...xn=1,
then x1+x2+x3+...+xn=n iff x1=x2=x3=...=xn=1
I tried to prove by induction. While the "if" direction is obvious, but I am kind of stuck in ...
0
votes
0
answers
79
views
Prove $ \frac{(c-a)^{2}}{6c} \le \frac{a+b+c}{3} - \frac{3}{ 1/a + 1/b+ 1/c}$
Given real numbers $c \ge b \ge a>0$, prove that
$$ \frac{(c-a)^{2}}{6c} \le \frac{a+b+c}{3} - \frac{3}{ \frac{1}{a} + \frac{1}{b} + \frac{1}{c}}$$
*using well-known inequality
Other solution ...
- 4,590