# Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities. Do not use this tag just because an inequality appears somewhere in your question.

30,543 questions
Filter by
Sorted by
Tagged with
5 views

### Ways of finding upper bounds of hypergeometric functions

I realized that I don't really know any good ways of finding bounds of hypergeometric functions after ${}_2 F_1$. For example, numerical evaluation convinced me that the generalized hypergeometric ...
25 views

• 5,200
33 views

45 views

• 752
357 views

### "Peeling Technique" in Probability

So I am reading "Bandit Algorithms" by Lattimore wherein for one of the proofs he uses a technique called as "Peeling Device" which he says is a widely used tool in probability. I ...
• 75
1 vote
33 views

• 579
342 views

### Show that $\frac{a^2}{b^2-2b+2024}+ \frac{b^2}{c^2-2c+2024}+ \frac{c^2}{a^2-2a+2024} \geq \frac{3}{2023}$

Let the real numbers $a,b,c \in \mathbb{R}$ with $a+b+c=3$. Show that: $\frac{a^2}{b^2-2b+2024}+ \frac{b^2}{c^2-2c+2024}+ \frac{c^2}{a^2-2a+2024} \geq \frac{3}{2023}$. My idea: First of all, I thought ...
• 752
59 views

### $a+b\sqrt{2}>1, a^2-2b^2=\pm 1 (a,b\in \mathbb{Z}) \implies a+b\sqrt {2}\geq 1+\sqrt {2}$

$\textbf{Example}:$ Let $K=\mathbb{Q}(\sqrt 2)$. We claim that $1+\sqrt 2$ is the fundamental unit of $K$. Easy to show that $N(1+\sqrt 2)=-1$ and thus a unit. Remain to show that if $v>1$ is any ...
• 1,541
1 vote
62 views

### Reference for $\sum_{m=1}^p\sum_{n=1}^q\frac2{\cos(2m\pi/p)+\cos(2n\pi/q)}\le pq(|p-q|+1)$ with coprime odd positive integers $p$ and $q$? [closed]

I am having trouble with the following problem that I found in this Art of Problem Solving post, and I would like some help to find a reference for it. Let $p$ and $q$ be coprime odd positive integers....
• 37
1 vote
72 views

### Prove that $\displaystyle \sum\limits_{i=1}^3\sqrt{ \sum\limits_{j=1}^3a_{ji}^2}\leq\sqrt{2}f(a_{11},\cdots,a_{33})$

For a $3\times3$ matrix $A=(a_{ij})$, let \begin{aligned}&f(a_{11},a_{21},a_{31},a_{12},a_{22},a_{32},a_{13},a_{23},a_{33})\\=&\text{max}\{|a_{11}+a_{21}+a_{31}|+|a_{12}+a_{22}+a_{32}|+|a_{13}+...
• 701
77 views

### Show that : $\sqrt{[x]\cdot \{x\}} +\sqrt{x \cdot \{x\}} + \sqrt{[x]\cdot x} \leq 2x$

Show that for any positive real number $x$ the inequality holds: $\sqrt{[x]\cdot \{x\}} +\sqrt{x \cdot \{x\}} + \sqrt{[x]\cdot x} \leq 2x$ where by $[a], \{a\}$ we mean the whole par and fractional ...
• 752
197 views

### Finding the integer part of a sum

I want to find the integer part of $$\sum_{n=1}^{10^9}\frac{1}{n^{2/3}}=S$$ I know there is a way using integration but I tried using a different approach. I saw this approach with square roots but I ...
• 107
1 vote
21 views

### Upper bound for distribution function for variable with zero expectation. [duplicate]

A problem from final Year 1 probability exam. Is it true for any random variable $Y$ s.t. $E[Y]=0$ and $E[Y^2]<\infty$ that: $P(Y>x)\leq\frac{E[Y^2]}{E[Y^2]+x}$ ? I thought we can rewrite it ...
• 124
1 vote
76 views

### Find min and max of $P = (|a − b| + 3)(|b − c| + 3)(|c − a| + 3)$

For $a,b,c \in \mathbb{R}$, $a^2 + b^2 + c^2 ⩽ 2$ Find min and max value of $P = (|a − b| + 3)(|b − c| + 3)(|c − a| + 3)$ I don't understand how to find min and max value of an absolute value sign. ...
17 views

• 1