# 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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### 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:/...
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### 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 ...
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### 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 $$\dfrac{x^2}{\left(x^2+1\right)^2}+\dfrac{y^2}{\left(y^2+1\right)^2}+\dfrac{z^2}{\left(z^2+1\right)^2}+\dfrac{u^2}{\left(u^2+1\right)^2} \leq \dfrac{16}{25}$$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ... 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 ...
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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 ...
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### 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 $n$...
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### 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 ...
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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&...
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### 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 ...
1 vote
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### 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. ...
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### 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 ...
1 vote
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### 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:...
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