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.

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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 ...
marty cohen's user avatar
4 votes
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139 views

Do the $4$ Pythagorean means divide into two algebraic dualities?

I’m looking to understand the relationship between rectangular product & mean square operations -- or, equivalently, between their root operations, the geometric & quadratic means. Here’s the ...
jasper's user avatar
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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 ...
Andres's user avatar
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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 ...
ziggurism's user avatar
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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 ...
Manoj Kumar's user avatar
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115 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{\...
TaD's user avatar
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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 ...
Russell Ng's user avatar
3 votes
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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=\...
Pck Tsp's user avatar
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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\...
Miss and Mister cassoulet char's user avatar
3 votes
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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 ...
Book Of Flames's user avatar
3 votes
2 answers
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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 ...
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$ \frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{a+c}=2 $ then $ \sum_{cyc}\sqrt{ab} \geq \frac{3}{2}$

Prove that if $$ \frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{a+c}=2 $$ then $$\sum_{cyc} \sqrt{ab} \geq \frac{3}{2}$$ I tried to use AM-GM inequality and Titu's lemma and was able to show that $a+b+c\...
roro roro's user avatar
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Basic Inequality problem related to AM-GM Inequalities

I am looking to prove $$ \frac{ab}{a⁵+b⁵+ab} + \frac{bc}{b⁵+c⁵+bc} + \frac{ac}{a⁵+c⁵+ac} \le1 $$ Where $ a,b,c $ are positive reals with $ abc = 1 $ My initial thought was to apply AM-GM to get $$ a⁵ +...
Aniket Kumar's user avatar
2 votes
0 answers
94 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,...
Ioraboi's user avatar
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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 ...
polite proofs's user avatar
2 votes
0 answers
99 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 ...
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2 votes
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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^{\...
Miss and Mister cassoulet char's user avatar
2 votes
0 answers
79 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 $\...
ArtOfProblemSolving's user avatar
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 = ...
Satya Prakash's user avatar
2 votes
1 answer
104 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 ...
SihOASHoihd's user avatar
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2 votes
1 answer
227 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 ...
Mathejunior's user avatar
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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 ...
user168826's user avatar
1 vote
3 answers
94 views

Regarding the specifity of the A.M.-G.M. inequality in finding maximum and minimum in Number Theory

Today, my math teacher solved a problem which asked to find the maximum value of the expression $x^2y^3$ when $x$ and $y$ are related as $3x+4y=5$. It was solved using the classic A.M.-G.M. inequality ...
BlackKnight23's user avatar
1 vote
0 answers
160 views

Brazilian Mathematical Olympiad 2023, Level U, Problem 3

Question Prove that there exists a constant $C > 0$ such that, for any integers $m, n$ with $n \geq m > 1$ and any real number $x > 1$,$$\sum_{k=m}^{n}\sqrt[k]{x} \leq C\bigg(\frac{m^2 \cdot \...
Martin.s's user avatar
1 vote
0 answers
59 views

Can we prove Am-GM Inequality using Karamata’s Inequality?

Recently I was trying to prove AM-GM Inequality using several different methods. When it came to Karamata’s Inequality, the proof that flashed through my mind was so simple and amazing, but it came ...
Arachnephob1a's user avatar
1 vote
0 answers
186 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 ...
aarbee's user avatar
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1 vote
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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{$*$...
KHOOS's user avatar
  • 407
1 vote
0 answers
199 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 &...
MathStackExchange's user avatar
1 vote
0 answers
97 views

Prove $ \frac{(c-a)^{2}}{6c} \le \frac{a+b+c}{3} - \frac{3}{ \frac{1}{a} + \frac{1}{b}+ \frac{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 ...
Redsbefall's user avatar
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1 vote
1 answer
84 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})^...
vekin pirna's user avatar
1 vote
0 answers
41 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 ...
Equalityisillusion's user avatar
1 vote
1 answer
80 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 ...
sunny's user avatar
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1 vote
0 answers
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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{...
MathMagician's user avatar
1 vote
1 answer
117 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 ...
AW23's user avatar
  • 77
1 vote
1 answer
54 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}{...
Cookie's user avatar
  • 437
1 vote
0 answers
220 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 $n$...
Zifeng Zhang's user avatar
1 vote
0 answers
76 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 ...
marty cohen's user avatar
1 vote
0 answers
355 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&...
geodude's user avatar
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1 vote
0 answers
50 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 ...
charu rama's user avatar
1 vote
0 answers
33 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. ...
Adam Rubinson's user avatar
1 vote
0 answers
39 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:...
taciturno's user avatar
  • 480
1 vote
0 answers
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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 ...
Miss and Mister cassoulet char's user avatar
0 votes
0 answers
72 views

Prove that $\prod_{i=1}^n(1-x_i)\ge 1/2$.

Let $x_1,x_2,\ldots, x_n\in \Bbb R^+\cup \{0\}$ such that $x_1+\cdots +x_n\le 1/2$. Prove that $\displaystyle \prod_{i=1}^n(1-x_i)\ge 1/2$. It is easy to prove it by induction but I have a doubt since ...
James A.'s user avatar
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0 answers
57 views

Reversing AM-GM Inequality When We Have Bounded Variable

Suppose $x_1,x_2,\dots,x_n$ are positive numbers. Define $A = \frac{1}{n}\sum_{i=1}^n x_i$, $G = (\Pi_{i=1}^n x_i)^{1/n}$. The well-known AM-GM inequality tells us that $$ A \geq G $$ Now suppose we ...
EggTart's user avatar
  • 375
0 votes
1 answer
134 views

How to prove $\frac{1}{\sqrt{a+b}}+\frac{1}{\sqrt{c+b}}+\frac{1}{\sqrt{a+c}}\ge 2+\sqrt{\frac{2}{3a+3b+3c-2}}$?

Question. Let $a,b,c\ge 0: ab+bc+ca=1.$ Prove that $$\color{black}{\frac{1}{\sqrt{a+b}}+\frac{1}{\sqrt{c+b}}+\frac{1}{\sqrt{a+c}}\ge 2+\sqrt{\frac{2}{3a+3b+3c-2}}. }$$ I've tried to use Jichen lemma ...
Anonymous's user avatar
0 votes
0 answers
40 views

How to prove this integral inequality $\int_{ - 1}^1 {{{\left| {g\left( s \right)} \right|}^2}ds}$

Question Let $g\in C_0^\infty((-1,1))$.Prove $\forall t\in (-1,1)$,$${g^4}\left( t \right) \le 16\int_{ - 1}^1 {\left( {{{\left| {g'\left( s \right)} \right|}^2} - \frac{{{g^2}\left( s \right)}}{{4{{\...
Martin.s's user avatar
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0 answers
45 views

Proving Inequality Involving Sums and Square Roots with Given Conditions

Question $$\text{Let } b_{i}\wedge a_{i}>0 \text{ where } i\in{1,2,3,\ldots,n} \nonumber , \sum_{i=1}^{n}(b_{i}) = \lambda \text{ then Prove that} \nonumber \frac{\lambda-(b_{1}+b_{2})}{(b_{1}+b_{...
Martin.s's user avatar
0 votes
0 answers
51 views

inequality about elementary symmetric polynomials with real variables

Fix $k\in \mathbb{N}$. Suppose $y_1,\ldots, y_{2k}$ are real numbers. Then $$\left(\sum_{1\le i_1<i_2<\cdots<i_k\le 2k} y_{i_1}y_{i_2}\cdots y_{i_k}\right)^2 \ge \binom{2k}{k}^2y_1y_2\cdots ...
Sayan's user avatar
  • 2,688
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0 answers
25 views

Homogeneous Inequality problem unsolved

Let ${x,y,z}$ be positive real numbers. Prove that $\sum\limits_{cyc}\frac{x^2}{yz+\frac{x^4}{y^2}+\frac{x^4}{z^2}} \leq 1$ Here’s my try: By A.M.-G.M. we know that $\frac{yz}{2}+\frac{yz}{2}+\frac{x^...
UWU11's user avatar
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0 answers
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Show that $\sqrt{\frac{a}{b+c}+ \frac{b}{c+a}}+ \sqrt{\frac{b}{c+a}+ \frac{c}{a+b}} + \sqrt{\frac{c}{a+b}+ \frac{a}{b+c}} \ge 3.$

Suppose that $a,b,c>0$, show that $$\sqrt{\frac{a}{b+c}+ \frac{b}{c+a}}+ \sqrt{\frac{b}{c+a}+ \frac{c}{a+b}} + \sqrt{\frac{c}{a+b}+ \frac{a}{b+c}} \ge 3.$$ Appreciate any advice!
Steven Lu's user avatar
  • 1,037