# Questions tagged [inequality]

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

4,506 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
2k views

### A Nice Problem In Additive Number Theory

$\color{red}{\mathbf{Problem\!:}}$ $n\geq3$ is a given positive integer, and $a_1 ,a_2, a_3, \ldots ,a_n$ are all given integers that aren't multiples of $n$ and $a_1 + \cdots + a_n$ is also not a ...
• 6,084
2k views

487 views

### Smallest $c$ such that $f'<cf$ holds for all $f$ such that $f,f',f'',f'''>0$ and $f''' \le f.$

Let $f: \mathbb{R} \to \mathbb{R}$ be a $C^3$ function such that $f,f',f'',f'''>0$ and $f''' \le f.$ What is the smallest $c$ such that we can guarantee $f'<cf$? Since $f(x)=e^x$ works, we must ...
• 5,154
662 views

### Bounding a polynomial from below

Let $\sigma >0$ be fixed. For even $k \in \mathbb{N} \cup \{0\}$, we consider the polynomial \varphi_k(x) = \sum_{j=0}^{k} (-1)^j {k \choose j} b_j \, x^{2j} \quad x \in (-1,1), \...
• 666
722 views

### A very general method for solving inequalities repaired

Yesterday, I asked a question about a very general method for solving equations I had found here. As it turned out, there were quite some problems with my method and I got a lot of good feedback. ...
• 8,361
339 views

### Ratio of product from one point and minimum distance

Given points $A_0,A_1,\ldots,A_n$ in the plane, let $m$ denote the minimum distance among any two points. What is the minimum value of $$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}?$$ ...
• 2,653
666 views

• 131
332 views

• 313
593 views

• 681
492 views

### On the Abstract Concreteness Method (bka $ABC-$Method).

I was reading Zdravko Cvetkovski's excellent book Inequalities: Theorems, Techniques, and selected problems, when I arrived at the $16$th chapter: the $ABC-$Method. I had some questions related to ...
• 8,621
478 views

### How to determine the minimal constant $\lambda = \lambda(n,k)$

Given fixed positive integers $n,k$, determine the minimal constant $\lambda = \lambda(n,k)$ for which the following inequality holds for any $a_1,a_2,...,a_n>0$ (taking indices mod $n$ if required)...
• 93.4k
463 views

• 93.4k
296 views

• 1,356
354 views

• 121
160 views

262 views

### Prove $\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$

Let $f\in C^1([0,1])$. Prove the following: $$\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$$ and $$|f(1/2)|\le\int_0^1(|f(t)|+\frac12|f'(t)|)dt$$ Note that $e^{-x}(e^xf(x))'=f(x)+f'(x)$. ...
• 4,976
335 views

### Close power towers

In another question I gave some ways to determine which of two power towers is larger, but my answer there is incomplete because it doesn't handle the case where two towers are very close at each ...
• 10.9k