# Tagged Questions

Questions on proving, manipulating and applying inequalities.

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### A conditional inequality which itself implies a sharper version of it [duplicate]

Problem: Given that $m, n$ are positive integers such that $\sqrt{7} -\frac{m}{n} > 0$. Then show that $\sqrt{7}-\frac{m}{n} > \frac{1}{mn}$. I have failed to do this fascinating problem. My ...
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### Poincaré-like inequality

Let $\Omega\subset\mathbb{R}^3$ be an open bounded set. Let $\partial\Omega=\Gamma^1\cup\Gamma^2$, with $\Gamma^1\cap\Gamma^2=\emptyset$. We denote as $\Gamma^1_j$, $j=1,\dots,p_{\Gamma^1}+1$, the ...
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### An inequality involving $\frac{x^3+y^3+z^3}{(x+y+z)(x^2+y^2+z^2)}$

$$\frac{x^3+y^3+z^3}{(x+y+z)(x^2+y^2+z^2)}$$ Let $(x, y, z)$ be non-negative real numbers such that $x^2+y^2+z^2=2(xy+yz+zx)$. Question: Find the maximum value of the expression above. ...
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### $\mathbb{P}[X_1(k^\ast)] \leq \left( \frac{e}{k^\ast} \right)^{k^\ast} \frac{1}{1-e/k^\ast} \leq n^{-2}$ inequality is used to prove the theorem

In the book Randomized Algorithms from Motwani and Raghavan, it is stated in page 44 that $$\mathbb{P}[X_1(k^\ast)] \leq \left( \frac{e}{k^\ast} \right)^{k^\ast} \frac{1}{1-e/k^\ast} \leq n^{-2}.$$ ...
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### Explanation of this integral identity in the proof of Wirtinger's inequality from Hardy-Littlewood-Polya

I report the following excerpt from the book "Inequalities" by Hardy-Littlewood-Polya, page 184, where Wirtinger inequality is proven using variational methods. I'm trying to understand what is the ...
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### Model linearly: What products to make, how much to make and in what plants to make them?

A company wants to make 3 new products for the upcoming week. We are given that: Each product can be made in 1 of 2 plants. At most 2 of the 3 new products should be chosen to be made. Only 1 of ...
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### Any advice on how to tackle this inequality? ($x^{a}e^{x+c}\leq b$ )

How might one go about solving the inequality: $x^{a}e^{x+c}\leq b$ where $a,b,c$ are arbitrary constants ($b\geq 0$ and $a\neq0$) for $x$. My first place would be to try and get all of the ...
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### What are some good books on algebraic inequalities?

By algebraic inequalities I mean inequalities like Cauchy's inequality, the AM-GM inequality etc. I need it for the International Mathematics Olympiad (IMO), so I hope I can find some books that ...
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### How do I show that $\frac a{1 - a^2} + \frac b{1 - b^2} + \frac c{1 - c^2} \ge \frac {3 \sqrt 3}2$

For $0 \lt a, b, c \lt 1$, if $ab + bc + ca = 1$, show that $$\frac a{1 - a^2} + \frac b{1 - b^2} + \frac c{1 - c^2} \ge \frac {3 \sqrt 3}2.$$ I want to use trigonometric substitution: For the ...
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### Name for inequality about sums of exponents with same base

Is there a name for the following inequality regarding sums of exponents which share a base? $$\text{For all integers b \geq 2, n \geq 1,} \\ \sum_{i=0}^{n-1}{b^i} < b^n$$
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### Model linearly: Determine amount of units for production

A company produces 2 products in a week. Let $x_i$ denote the number of units of product $i$ to produce. Each product requires liters of Chemical X to make. Info is given below: \begin{array}{|c|c|} ...
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### Expectation inequality

How can I prove that $$\mathbb{E} [X^2] \geq \mathbb{E} [|X|]^2$$ This resembles a lot the Cauchy-Schwarz inequality but I'm unable ti prove it with the usual method (i.e. when there are two random ...
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### How to prove $({\log_2 x})^{n+1} \le x^n$

I want to show that $({\log_2 x})^{n+1} \le x^n$ when $n \ge 1$ and $x \ge 1$. I know that ${\log_2 x}$ can be shown to be $\lt x$ with: $x \lt 2^x$ $\log_2 x \lt x$ and obviously adding the same ...
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### An inequality to find the range of an unknown coefficient

Find the range of $a$ such that $$a(x_1^2+x_2^2+x_3^2)+2x_1x_2+2x_2x_3+2x_1x_3 \geq 0, x_i\in \mathbb{R}$$ I tried to use Cauchy Inequality but it seems not...
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### Under what conditions the following inequality holds?

I have the following the inequality holds? $A+B>w^{1/m}A+w^{1-1/m}B$ where $A>0, B>0$, $w$ and $m$ are positive integer number satisfying $1\leq w,m<\infty$. My question is under what ...
### How do I prove that $\left | \sum_{j=1}^n a_j \right |^2 + \left | \sum_{j=1}^n (-1)^j a_j \right |^2 \le (n+2) \sum_{j=1}^n a_j^2$?
For any $a_j \in \Bbb R, \, j = 1, 2, \cdots, n$, one has the bound $$\left | \sum_{j = 1}^n a_j \right |^2 + \left | \sum_{j = 1}^n (-1)^j a_j \right |^2 \le (n + 2) \sum_{j =1}^n a_j^2.$$ This is ...