$f: D\to D$ $(D = \{|z| &lt; 1\})$, is a holomorphic function. How to proof  $$ \left|\frac{f(z_1)-f(z_2)}{z_1-z_2}\right| \leq \frac1{1-r^2}, \quad\forall |z_1|,|z_2| \leq r $$ I try using the Schwarz-Pick theorem which gives, $$ \left|\frac{f(z_1)-f(z_2)}{1-\overline{f(z_1)}f(z_2)}\right| \leq \left|\frac{z_1-z_2}{1-\bar z_1 z_2}\right| $$ then it gives, $$ \left|\frac{f(z_1)-f(z_2)}{z_1-z_2}\right| &lt; \frac2{1-r^2} $$ Thanks a lot.
I am trying to prove that:  Let $f$ be analytic in the closed unit disc with $f(z)\leq 1$ for $|z|\leq1$.  For a given $0&lt;r&lt;1$ and $|z_1|,|z_2|\leq r$, $z_1\neq z_2$ then,   $$\left|\frac{f(z_2)-f(z_1)}{z_2-z_1}\right|\leq \frac{1}{(1-r)^2}.$$  My attempt is: part a  of the question asks to prove $$\left|\sum_k^\infty a_nz^n\right|\leq \frac{ |z^k|}{1-|z|}.$$   So, \begin{eqnarray*}  \left|\frac{f(z_2)-f(z_1)}{z_2-z_1}\right|&amp;=&amp; \frac{|\sum_1^\infty a_nz_2^n-\sum_1^\infty a_nz_1^n|}{|z_2-z_1|}\\  &amp;\leq&amp; \frac{\frac{ |z_1 |}{1-|z_1|}+\frac{ |z _2|}{1-|z_2|}}{|z_2-z_1|}\\  &amp;\leq&amp;\frac{1}{(1-r)^2} \frac{   |z_2| ({1-|z_1|})+  |z _1| (1-|z_2|) }{|z_2-z_1|}\\ \end{eqnarray*} But could not prove that the last term is controlled by 1.
Java: splitting a comma-separated string but ignoring commas in quotes What can I use to typeset MATLAB code in my document? Checking for valid floating point number $$|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$$   I have tried: $$|a\sin x+b \cos x|\leq |a+b|\leq \sqrt{a^2+b^2}$$  enough to prove: $$|a+b|\leq \sqrt{a^2+b^2}$$  But I can find how to continue from here In Java, are there clear rules on when to use each of access modifiers, namely the default (package private), public, protected and private, while making class and interface and dealing with inheritance? How to store lots of small text in a SQL Server database? What exactly do quotation marks around the table name do? I thought that the quotation mark (") was simply a type of grouping marker but I'm debugging some NHibernate code and notice that while  SELECT * FROM site WHERE site_id = 3;   Works fine  SELECT * FROM "site" WHERE site_id = 3;   fails with a table or view does not exist error.  What gives? Finding the internal name and display name for a list column I am using SharePoint Server 2007 Enterprise with Windows Server 2008 Enterprise, and I am using publishing portal template. I heard for SharePoint list/list column, it has an internal name which could be used for programming (as identifier for list/list column), and also has a display name (which is showed for end user for list name or list column name).  How to find and set such internal name and display name for list/list column? Join table depending on value of column in main table How can I display keyboard shortcuts as I press them? I am encountering two definitions of the special orthogonal lie algebra, and I would like to know if they are equivalent, and if there are advantages to working with one over the other.  If we begin with an $n$-dimensional vector space $V$ over a field $k$ and a chosen basis, we can define a bilinear form on $V$ by a matrix $S\in M_n(k)$, ie, let $\langle v,w\rangle=v^tSw$ for all $v,w\in V$.  Now $g\in GL_n(k)$ preserves the form ($\langle g(v),g(w)\rangle=\langle v,w\rangle$) if and only if $g^tSg=S$, so all such $g$ form a linear algebraic group $G$.  The tangent space at the identity of $G$ will be contained in that of $GL_n(k)$, so $T_eG\subset T_eGL_n(k)=M_n(k)$, and in fact, $T_eG=\{B\in M_n(k)\mid B^tS+SB=0\}$.  $T_eG$ becomes a lie algebra, $Lie(G)$, if we define the bracket to be the commutator of two matrices.  Now, if $S=I_n$, it follows that $G=O_n(k)$ is the orthogonal group of matrices satisfying $g^tg=I_n$, and $Lie(G)=\mathfrak{so}_n$ is the lie algebra of antisymmetric matrices.  In Humphrey's Introduction to Lie Algebras and Representation Theory, he defines $\mathfrak{so}_n$ to be all matrices $B$ satisyfing $B^tS+SB=0$, where $$ S=\begin{pmatrix} 1&amp;0&amp;0\\ 0&amp;O&amp;I_l\\ 0&amp;I_l&amp;O \end{pmatrix} \hspace{.5in}\text{or}\hspace{.5in} S=\begin{pmatrix} O&amp;I_l\\ I_l&amp;O \end{pmatrix} $$ depending on the parity of $n$.  The matrices obtained in this way are not antisymmetric, nor is the group $G$ preserving the form defined by $S$ the orthogonal group $O_n(k)$.  Are the two groups obtained from considering different $S$ isomorphic?  Are the two lie algebras isomorphic?  If so, why would one prefer one form to the other? Avoiding errors due to unexpanded asterisk In bash I often use for-loops such as the following  for file in *.type; do    sommecommand "$file";  done;   to perform an operation for all files matching *.type. If no file with this ending is found in the working directories the asterisk is not expanded and usually I will get an error message saying that somecommand didn't find the file. I can immediately think of several ways to avoid this error. But adding a conditional does not seem to be very elegant. Is there a short and clean way to achieve this? Travelling to Italy without my spouse who is a UK citizen I am India citizen married to a UK national who has been issued a Schengen Visa from the Italian consulate as a spouse of a UK national. My husband was supposed to travel with me but now can’t. We have decided that I should still pursue the trip without him as I will be joining our friends there. Please can someone let me know if I will still be allowed to travel to the Schengen region with a valid Schengen Visa issued as a spouse of a EEA national but without my husband joining me there in person ? How to view the system console message after Blender crashes? I am testing a script in Blender game engine. I can view the helpful message on the external system console when the script throws exception. Sometimes the bugs of the script cause the Blender to crash and close the console immediately.    Is there any way to view the message after the Blender crashes? How to write the message to a text file for later viewing? Any suggestion? So I have this word problem and I’m a bit confused about it. I have the answer and explanation but I still don’t understand: The half-life of carbon-14 is approximately 5730 years, while the half-life of carbon-12 is essentially infinite. If the ratio of carbon-14 to carbon-12 in a certain sample is 25% less than the normal ratio in nature, how old is the sample?     A. Less than 5730 years      B. Approximately 5730 years      C. Significantly greater than 5730 years, but less than 11460 years      D. Approximately 11460 years  Correct Answer: A Explanation: Because the half-life of carbon-12 is essentially infinite, a 25 percent decrease in the ratio of carbon-14 to carbon-12 means the same as a 25 percent decrease in the amount of carbon-14. If less than half of the carbon-14 has deteriorated, then less than one half-life has elapsed. Therefore, the sample is less than 5730 years old. Be careful with the wording here—the question states that the ratio is 25% less than the ratio in nature, not 25% of the ratio in nature, which would correspond to choice (D). How is the ratio of the carbon isotopes relevant to half-lives? What is the purpose of saying the half-life of an isotope is infinite? What is meant by “the normal ratio in nature”? Just based on the answer, it seems like the question said “25% of a carbon sample decayed, how old is this sample?” Obviously it’s younger than 5,730 years (the time for its first half life) because if only 25% decayed that means half a half-life has passed. I don’t see how isotopes and ratios would change this problem at all. Edit: Every answer here was very informative and helpful. It was tough to pick a best answer. I picked Cosma’s because it clicked with me. If you’re reading this and want to understand radiocarbon dating, look at every answer because there is some useful information in every answer Show that $a_{n+1}= 1 + \frac{1}{a_n}$ converges. Let $\{a_n\}$ be defined by $a_1 =1  $ and $a_{n+1} = 1 + {\dfrac{1}{a_n}}$ with $n \in N$. Show that $a_{n+1}= 1 + {\dfrac{1}{a_n}}$ converges.  I know the limit but how can I show that is a Cauchy sequence or that this sequence converges. When is $\mathbb{F}_p[x]/(x^2-2)\simeq\mathbb{F}_p[x]/(x^2-3)$ for small primes? How do I clear keyboard/autocorrect history on the Galaxy s4? I let my younger cousins play on my S4 from time to time, but now I have found that all the words they have used are the first to come up when I use my phone to text. It has gotten quite irritating. Samsung seems to have very hard to find settings compared to my old Sony Ericsson so I cannot find where to get rid of the word history typed onto my phone's keyboard. I'm a teacher and I teach at/in school OR at/in a school There are a lot of posts on this topic, but I can't find any answers to my question.  I came across the following two definitions of 'school':        [uncountable] (used without the or a) the process of learning in a   school; the time during your life when you go to a school (British   English)   [uncountable] (used without the or a) the time during the day when   children are working in a school      Source:   I know there are better ways to convey the message, but which sentence would a teacher utter if he was a school teacher teaching students, adults, etc.? What about a janitor, director, and secretary?     I teach at/in school OR I teach at/in a school   The above definitions concern students, and not working individuals like teachers or janitors.  I would appreciate if you also stated whether you're American, Canadian, British, etc. How do I change my username?