# Tagged Questions

Questions on proving, manipulating and applying inequalities.

44 views

17 views

### Can I presume that this inequality is a good aproximation for a divisor function?

I've used the Lemma 7.9 from page 73 from Krizek, Luca and Somer, 17 Lectures on Fermat Numbers From Number Theory to Geometry Springer CMS (2001) (you can see this page as a Google Book, type here ...
85 views

In another post an inequality referred to as "Etemadi's Inequality" is mentioned twice - in the original post as well as in the answer. However, the contexts of usage are such as to raise the question ...
60 views

21 views

### $|\mathcal{R}((2a+ib)^{2n+1})|\neq b$ for coprime $2a,b$ and $n>1$

Assume $n>1$ is natural and set $f_n(a,b):=\mathcal{R}((2a+ib)^{2n+1})$ Prove that for every coprime pair $2a,b\in\mathbb{Z}$: $|f_n(a,b)|>b$. Note that we have $b|f_n(a,b)$ so the only thing ...
45 views

### Maximum value of $\sum_{i\neq j}a_ia_j$ subject to $\sum_{i=1}^n a_i=1$

Find the maximum value of $\sum_{i\neq j }a_ia_j$ subject to $\sum_{i=1}^n a_i=1$. Here, $a_i\in\mathbb R$, for all $i$. Can I take $a_i>0$ for all $i$? If yes, then I can actually use AM-GM and ...
23 views

68 views

### A **proof** for $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2}$ [duplicate]

I need a proof for the inequality: $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2}$ for all natural numbers $t \geq 2$. For $t=2$ both sides are equal. Can someone find a proof for all $t$? maybe ...
64 views

### Is this number positive?

Let $(a_{ij})$ be a collection of non-negative numbers indexed by integers $1\le i,j \le N$ where $N$ is some fixed integer. Let $(c_{ij})$ be another collection of real numbers also indexed by ...
105 views

### A proof for the inequality $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2}$ for all $t \geq 2$

I'm struggling with proving the following inequality: $$\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2}$$ for all $t \geq 2$. I think it is monotonic non-increasing in $t$, which would suffice. ...
44 views

### How prove this inequality with $3(x^2-x+1)(y^2-y+1)(z^2-z+1)$

Let $x,y,z$ be real numbers, prove that $$3(x^2-x+1)(y^2-y+1)(z^2-z+1)\ge (xyz)^2+x^2(y+z)+y^2(z+x)+z^2(x+y)-5xyz+1$$ with equality if $x=0,y=z=1$ .
68 views

25 views

### Inequality involving the length of a vector and its components

Given a vector $\overrightarrow{v}$ of length $|\overrightarrow{v}|$ with components $x,y,z\gt 0$ how to prove the following inequality? $$|\overrightarrow{v}|^2\ge\sqrt 2(xy+yz)$$ Thanks
108 views

51 views

### Solving the Inequality $\frac{14x}{x+1}<\frac{9x-30}{x-4}$

The question says to find all the integral values of x for which the inequality holds. the question is $$\frac{14x}{x+1}<\frac{9x-30}{x-4}$$ My Solution \begin{align} & \frac{14x}{x+1} < ...
42 views

32 views

### Using induction on modified inequalities.

Here's the original problem: Prove by induction that $\left(\frac{1}{2}\right) \left(\frac{3}{4}\right) \cdots \left(\frac{2n-1}{2n} \right) \leq \frac{1}{\sqrt{n+1}}$ for all $n \in \mathbb{N}$. ...
### Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$
For every real positive n prove that $\sqrt{4n+1}<\sqrt{n}+\sqrt{n+1}<\sqrt{4n+2}$.Hence,or otherwise prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$. Wher [x] denotes the greatest integer not ...