Articles of means

Prove an algorithm for logarithmic mean $\lim_{n \to \infty} a_n=\lim_{n \to \infty} b_n=\frac{a_0-b_0}{\ln a_0-\ln b_0}$

Take: $$a_0=x,~~~~b_0=y$$ $$a_{n+1}=\frac{a_n+\sqrt{a_nb_n}}{2},~~~~b_{n+1}=\frac{b_n+\sqrt{a_nb_n}}{2}$$ Then we obtain as a limit the logarithmic mean of $x,y$: $$\lim_{n \to \infty} a_n=\lim_{n \to \infty} b_n=\frac{x-y}{\ln x-\ln y}$$ I don’t know how to prove this. But I do know that numerically it fits really well. In fact, the best approximation is obtained if we take geometric mean of $a_n,b_n$: $$x=5,~~~~y=3$$ […]

Mean and variance of the order statistics of a discrete uniform sample without replacement

In answering calculate the mean and variance of the highest number drawn on lottery based on the lowest number drawn, I couldn’t find the mean and variance of the order statistics of a discrete uniform sample without replacement anywhere, so I figured I’d derive them here for future reference. Draw $k$ distinct numbers uniformly from […]

Is it true that $\limsup \phi\le\limsup a_n$, where $\phi=\frac{a_1+…+a_n}n$?

This question already has an answer here: If $\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$ then $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$ 2 answers

Computing square roots with arithmetic-harmonic mean

We know that if we iterate arithmetic and harmonic means of two numbers, we get their geometric mean. So, basically if we need to compute the square root of $x$: $$\sqrt{x}=\sqrt{1 \cdot x}=AHM(1,x)$$ $$a_0=1,~~~~b_0=x$$ $$a_{n+1}=\frac{a_n+b_n}{2},~~~~~b_{n+1}=\frac{2a_nb_n}{a_n+b_n}=\frac{a_nb_n}{a_{n+1}}$$ As far as I know, this expression will converge for any real positive $x$. See this and this question for […]

Extending the ordered sequence of 'three-number means' beyond AM, GM and HM

I want to create an ordered sequence of various ‘three-number means‘ with as many different elements in it as possible. So far I’ve got $12$ ($8$ unusual ones are highlighted): $$\sqrt{\frac{x^2+y^2+z^2}{3}} \geq \color{blue}{ \frac{\sqrt{x^2+y^2}+\sqrt{y^2+z^2}+\sqrt{z^2+x^2}}{3 \sqrt{2}}} \geq $$ $$\geq \color{blue}{\frac{\sqrt{(x+y)^2+(y+z)^2+(z+x)^2}}{2 \sqrt{3}} } \geq \frac{x+y+z}{3} \geq $$ $$ \geq \color{blue}{ \frac{\sqrt[3]{(x+y)(y+z)(z+x)}}{2} } \geq \color{blue}{\sqrt{\frac{xy+yz+zx}{3}}} \geq $$ $$\geq […]

Closed form for the integral $\int_0^\infty \frac{dx}{\sqrt{(x+a)(x+b)(x+c)(x+d)}}$

Let’s consider the function defined by the integral: $$R(a,b,c,d)=\int_0^\infty \frac{dx}{\sqrt{(x+a)(x+b)(x+c)(x+d)}}$$ I’m interested in the case $a,b,c,d \in \mathbb{R}^+$. Obviously, the function is symmetric in all four parameters. This function has some really nice properties. $$R(ka,kb,kc,kd)=\frac{1}{k} R(a,b,c,d)$$ Thus: $$R(a,a,a,a)=\frac{1}{a}$$ Moreover: $$R(a,a,b,b)=\frac{\ln a-\ln b}{a-b}$$ This is the reciprocal of the logarithmic mean of the numbers $a$ and […]

Closed form for the limit of the iterated sequence $a_{n+1}=\frac{\sqrt{(a_n+b_n)(a_n+c_n)}}{2}$

Is there a general closed form or the integral representation for the limit of the sequence: $$a_{n+1}=\frac{\sqrt{(a_n+b_n)(a_n+c_n)}}{2} \\ b_{n+1}=\frac{\sqrt{(b_n+a_n)(b_n+c_n)}}{2} \\ c_{n+1}=\frac{\sqrt{(c_n+a_n)(c_n+b_n)}}{2}$$ in terms of $a_0,b_0,c_0$? $$L(a_0,b_0,c_0)=\lim_{n \to \infty}a_n=\lim_{n \to \infty}b_n=\lim_{n \to \infty}c_n$$ For the most simple case $a_0=b_0$ we have some interesting closed forms in terms of inverse hyperbolic or trigomonetric functions: $$L(1,1,\sqrt{2})=\frac{1}{\ln(1+\sqrt{2})}$$ $$L(1,1,1/\sqrt{2})=\frac{2 \sqrt{2}}{\pi}$$ […]

Question about Geometric-Harmonic Mean.

Define our Harmonic sequence for two numbers such that \begin{equation} a_{n+1} = \frac{2a_nb_n}{a_n + b_n} \end{equation} and our geometric sequence \begin{equation}b_{n+1} = \sqrt{a_nb_n} \end{equation} such that as $n \rightarrow \infty$ we tend towards the Geometric-Harmonic Mean. The arithmetic-geometric mean can be defined by the following two sequences. First compute the arithmetic mean of two numbers […]

Inequalities involving arithmetic, geometric and harmonic means

Let $A$, $G$ and $H$ denote the arithmetic, geometric and harmonic means of a set of $n$ values. It is well-known that $A$, $G$, and $H$ satisfy $$ A \ge G \ge H$$ regardless of the value $n$. Furthermore, for $n=2$ we have $$G^2 = AH$$ By coincidence I found the following result for $n=3$ […]

Iterated means $a_{n+1}=\sqrt{a_n \frac{b_n+c_n}{2}}$, $b_{n+1}$ and $c_{n+1}$ similar, closed form for general initial conditions?

For every nonnegative $(a_0,b_0,c_0)$, consider $$a_{n+1}=\sqrt{a_n \frac{b_n+c_n}{2}},\quad b_{n+1}=\sqrt{b_n \frac{c_n+a_n}{2}},\quad c_{n+1}=\sqrt{c_n \frac{a_n+b_n}{2}}$$ $$M(a_0,b_0,c_0)=\lim_{n \to \infty} a_n=\lim_{n \to \infty} b_n=\lim_{n \to \infty} c_n$$ This simple three term iterated mean gives several interesting closed forms for particular cases (confirmed to $14$ digits so far): $$M(1,1,2)=\frac{3^{3/4}}{\sqrt{\pi}}=1.28607413715749$$ $$M(1,1,\sqrt{2})=\frac{2}{\sqrt{\pi}}=1.12837916709551$$ $$M(1,1,\sqrt{3})=\frac{2^{3/4}}{\sqrt{\arccos(-1/3)}}=1.21670090936316$$ $$M(1,1,\sqrt{3}/2)=\frac{1}{\sqrt{\ln 3}}=0.95406458200000$$ These particular closed forms (if true) imply that the […]