Articles of inequality

SOS: Proof of the AM-GM inequality

“A basic strategy in tackling inequalities of few variables is to write things into sum of squares…” is a quote from an answer, intended as commenting the post entitled “Can this inequality proof be demystified?”. I’d like to know what the Sum Of Squares strategy yields for the AM-GM inequality with $n$ variables. In the […]

How to prove that ab $\le \int_{0}^{a} \phi(x) dx + \int_{0}^{b} \phi^{-1}(y) dy$

How to prove that if $\phi: [0, \infty) \to [0; \infty)$ continuous and strictly increasing function. Then there is an inverse $\phi^{-1}$ and for all positive $a, b:$ $$ab \le \int_{0}^{a} \phi(x) dx + \int_{0}^{b} \phi^{-1}(y) dy.$$ I think i can prove it graphically, but not sure if this prove is rigorous. Clearly that ab […]

Show that $\left(1+\frac{1}{n}\right)^{n}\geq \sum_{k=0}^n\left(\frac{1}{k!}\prod_{i=0}^{k-1}\left(1-\frac{i}{n}\right)\right)$

I have to prove that $$\left(1+\frac{1}{n}\right)^{n}\geq \sum_{k=0}^n\left(\frac{1}{k!}\prod_{i=0}^{k-1}\left(1-\frac{i}{n}\right)\right)$$

Find max: $\frac{a}{b+2a}+\frac{b}{c+2b}+\frac{c}{a+2c}$

This question already has an answer here: Olympiad inequality $\frac{a}{2a + b} + \frac{b}{2b + c} + \frac{c}{2c + a} \leq 1$. 2 answers

Positivness of the sum of $\frac{\sin(2k-1)x}{2k-1}$.

For $n\in \mathbb{N}$, $x\in (0,\pi)$. Prove that : $$f_n(x)=\sum_{k=1}^n \frac{\sin [(2k-1)x]}{2k-1} \geq 0.$$ I’ve tried to do it by differentiation : I Calculate $f_n'(x)$ (sum of $\cos(2k-1)x$) but It has many roots. So, I couldn’t go further.

Area of a cyclic polygon maximum when it is a regular polygon

My question: Let $n$ points $A_1, A_2,\ldots,A_n$ lie on given circle then show that $\operatorname{Area}(A_1A_2\cdots A_n)$ maximum when $A_1A_2\cdots A_n$ is an $n$-regular polygon.

If $a,b,c$ are positive integers, with $a^2+b^2-ab=c^2$ prove that $(a-b)(b-c)\le0$.

I have an inequality problem which is as follow: If $a,b,c$ are positive integers, with $a^2+b^2-ab=c^2$ prove that $(a-b)(b-c)\le0$. I am not so good in inequalities. So, please give me some hints so that I can proceed. Thanks.

If $a_1,\ldots,a_n>0$ and $a_1+\cdots+a_n<\frac{1}{2}$, then $(1+a_1)\cdots(1+a_n)<2$.

Assume that $a_1,a_2,\ldots,a_n>0$ and $a_1+a_2+\cdots+a_n<\frac{1}{2}$, and prove that $$(1+a_1)(1+a_2)\cdots(1+a_n)<2$$ I’ve tried Hölder’s inequality (the same result can easily be derived using AM-GM). I’ve found out that it’s sufficient to prove that $$ \left(\frac{2n+1}{2n}\right)^n<2. $$ (I’ve created this sign for myself to use informally while searching for a proof. Proving that one of the signs holds […]

$4x^{4} + 4y^{3} + 5x^{2} + y + 1\geq 12xy$ for all positive $x,y$

Prove this inequality: $$4x^4 + 4y^3 + 5x^2 + y + 1 \geq 12xy$$ if $x$ and $y$ are real and positive. Please, I am a beginner and have no idea how to solve this, so don’t use any strange theorems.

General Triangle Inequality, distance from a point to a set

I am trying with no luck to prove: Let (X,d) be a metric space and A a non-empty subset of X. For x,y in X, prove that d(x,A) < d(x,y) + d(y,A)