Articles of inequality

Is it true that $\mathbb{E}+|\mathbb{E}\rvert\geq\mathbb{E}\rvert]+\mathbb{E}\rvert]$?

I have two (real-valued) independent random variables $X,Y$, and a function $f\colon \mathbb{R}\times\mathbb{R}\to\mathbb{R}$. Is it true that $$ \mathbb{E}[\lvert f(X,Y) \rvert]+\lvert\mathbb{E}[ f(X,Y) ] \rvert \geq \mathbb{E}[\lvert \mathbb{E}[f(X,Y) \mid X]\rvert]+\mathbb{E}[\lvert \mathbb{E}[f(X,Y) \mid Y]\rvert] \tag{$\dagger$} $$ ? (If not, what is a counter-example?) It looks eerily simple and intuitive, but at that point I wouldn’t believe in […]

Conjecture about $A(z) = \lim b^{} ( c^{} (z) ) $

Let $b(z),c(z)$ be analytic on the strictly positive reals. Let $^{[*]}$ denote composition. Conjecture : If $A(z) = \lim b^{[n]} ( c^{[n]} (z) ) $ Such that 1) the limit ( $A(z)$ ) exists for all strictly positive real $z$. 2) the sequence $a_n(z) = b^{[n]} ( c^{[n] } (z) ) $ is bounded in […]

Help me prove this inequality :

How would I go about proving this? $$ \displaystyle\sum_{r=1}^{n} \left( 1 + \dfrac{1}{2r} \right)^{2r} \leq n \displaystyle\sum_{r=0}^{n+1} \displaystyle\binom{n+1}{r} \left( \dfrac{1}{n+1} \right)^{r}$$ Thank you! I’ve tried so many things. I’ve tried finding a series I could compare one of the series to but nada, I tried to change the LHS to a geometric series but that […]

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.