Articles of inequality

Proving (a part of) Hoeffding's lemma

Hoeffding lemma goes like this: *Let $X$ be a scalar variable taking values in an interval $[a,b]$. Then for any $t>0$ $$\mathbb{E} e^{tX}\leq e^{t\mathbb{E} (X)}(1+O(t^2\mathbb{V}(X)\exp(O(t(b-a)))).$$In particular $$\mathbb{E} e^{tX}\leq e^{t\mathbb{E} (X)}\exp(O(t^2(b-a)^2).$$(This version is taken from T. Taos book on Random matrices – it’s on pp. 61 Lemma 2.1.2 from the draft from his website) My question […]

Prove independence of a pairwise independent subsequence of independent events

Consider infinite independent coin tossing where $H_n = \{$nth coin is heads$\}$ for $n = 1, 2, …$. Let $$A_n = \bigcap_{i=1}^{\left \lfloor \log_2 n \right \rfloor} H_{n+i}$$ How do you show that $(B_n = A_{f(n)})$, where $f(n) = \left \lfloor n\log_2 n^2 \right \rfloor$, is an independent subsequence? I was able to show that […]

Absolute Value inequality help: $|x+1| \geq 3$

Find the solutions to the inequality: $$|x+1| \geq 3$$ I translate this as: which numbers are at least $3$ units from $1$? So, picturing a number line, I would place a filled in circle at the point $1$. The solutions would then be on the interval $(-\infty,-2] \cup [4,\infty)$. But this is wrong, because: Why […]

Find minimum of $a+b+c+\frac1a+\frac1b+\frac1c$ given that: $a+b+c\le \frac32$

Find minimum of $a+b+c+\frac1a+\frac1b+\frac1c$ given that: $a+b+c\le \frac32$ ($a,b,c$ are positive real numbers). There is a solution, which relies on guessing the minimum case happening at $a=b=c=\frac12$ and then applying AM-GM inequality,but what if one CANNOT guess that?!

Prove that $\frac{\tan{x}}{\tan{y}}>\frac{x}{y} : \forall (0<y<x<\frac{\pi}{2})$

Prove that $\frac{\tan{x}}{\tan{y}}>\frac{x}{y} : \forall (0<y<x<\frac{\pi}{2})$. My try, considering $f(t)=\frac{\tan{x}}{\tan{y}}-\frac{x}{y}$ and derivating it to see whether the function is increasing in the given interval. I should be sure that $\lim_{x,y\rightarrow0}\frac{\tan{x}}{\tan{y}}-\frac{x}{y}\geq0$ for the previous derivative check to be useful, which I’m not yet, but I’m assuming it’s $0$ since I’d say that since both $x,y$ approach […]

How prove this mathematical analysis by zorich from page 233

Let $f$ be twice differentiable on an interval $I$,Let $$M_{0}=\sup_{x\in I}{|f(x)|},M_{1}=\sup_{x\in I}{|f'(x)|},M_{2}=\sup_{x\in I}{|f”(x)|}$$ show that (a):$$M_{1}\le 2\sqrt{M_{0}M_{2}}$$ if the length of $I$ is not less than $2\sqrt{\dfrac{M_{0}}{M_{2}}}$ (b):the numbers $2$ and $\sqrt{2}$ (in part a) cannot be replaced by smaller numbers. My try:for part $(a)$ I can prove if the length of $I$ is not […]

System of Equations: any solutions at all?

I am looking for any complex number solutions to the system of equations: $$\begin{align} |a|^2+|b|^2+|c|^2&=\frac13 \\ \bar{a}b+a\bar{c}+\bar{b}c&=\frac16 (2+\sqrt{3}i). \end{align}$$ Note I put inequality in the tags as I imagine it is an inequality that shows that this has no solutions (as I suspect is the case). This is connected to my other question… I have […]

Proving $\frac{x}{x^2+1}\leq \arctan(x)$ for $x\in .$

How can I prove this inequality? $$\frac{x}{x^2+1}\leq \arctan(x) \, , x\in [0,1].$$ Thank you so much for tips! Sorry if this is just stupid.

How can I prove that the binomial coefficient ${n \choose k}$ is monotonically nondecreasing for $n \ge k$?

I want to prove that the binomial coefficient ${n \choose k}$ for $n \ge k$ is a monotonically nondecreasing sequence for a fixed $k$. How do I do this?

Inequality and Induction: $\prod_{i=1}^n\frac{2i-1}{2i}$ $<$ $\frac{1}{\sqrt{2n+1}}$

This question already has an answer here: Induction and convergence of an inequality: $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}\leq \frac{1}{\sqrt{2n+1}}$ 7 answers