$f:D \rightarrow \mathbb{C}$ holomorphic and $d:=\sup_{z,w\in D} |f(z)-f(w)|$. Prove that $d\geq 2|f'(0)|$ and is equal if and only if $f(z)=a_0+a_1z$
Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. I want to demonstrate that if $$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$$  then $f$ is linear.   I know this is a well-known result. Where can I find the proof ?
$A \subseteq \mathcal{R}(N)$ and given that (by inductive definition):   $N ∈ S$. If $a \in R$, then $R \setminus \{a\} \in A$.    I need to prove that for each $A \in S$, $N\setminus A$ is finite. How can I prove this formally? OrderBy and OrderByDescending are stable? How to construct semi direct products between Q and H There is a semi direct product if you go $\theta: Q \rightarrow Aut(H)$.  $H = C_{17}, Q = C_2$.  $Aut(H)\cong C_{16}$.  From here, how do I construct the semi direct products?   I said, in $C_{16}$, there are two elements of order that divide $2$, which are $1$ and $2$. Therefore there are two semi direct products. Also $17 \times 2 = 34$ tells me that in both semi direct products there will be $34$ elements.  How do I construct the semi direct products from here though? What compiler warning level do you recommend for different C/C++ compilers?  gcc and g++ will let you get away with a lot on the default level.  I find the best warning level for me is '-Wall'.  And I always try to remove fix the code for the warnings it generates.  (Even the silly ones about using parenthesis for logical precedence rules or to say I really mean 'if (x = y)')  What are your favorite levels for the different compilers, such as Sun CC, aCC (HPUX ?), Visual Studio, intel?  Edit:  I just wanted to point out that I don't use "-Werror" (but I do understand it's utility) on gcc/g++ because, I use:   #warning "this is a note to myself"   in a few places in my code.  Do all the compilers understand the #warning macro? Determinant of a block lower triangular matrix I'm trying to prove the following: Let $A$ be a  $k\times k$ matrix, let $D$ have size $n\times n$, and $C$ have size $n\times k$. Then,  $$\det\left(\begin{array}{cc} A&amp;0\\ C&amp;D \end{array}\right) = \det(A)\det(D).$$  Can I just say that $AD - 0C = AD$, and I'm done? Is reusing old code for a new assignment considered self plagiarism? How to protect yourself if you consider it to be, and a group partner does not? How to add CC to a webform email? I'm using Drupal 7.  Is it possible to add a CC to a webform email?  I was using   function theme_webform_mail_headers($variables) {   $headers = array(     'X-Mailer' =&gt; 'Drupal Webform (PHP/' . phpversion() . ')',   );   return $headers; }   changing it with      function MYTHEME_webform_mail_headers($variables) {       $headers = array( 'CC' =&gt; 'ccmeail@domain.com', 'X-Mailer' =&gt; 'Drupal Webform (PHP/'. phpversion() .')',  );       return $headers;     }   It does not work, I've cleared all the chaches and it does not add the CC. How could I add the CC?  Thanks in advance. yum install error I can't run a yum command on my server without getting this:   CRITICAL:yum.cli:Config Error: Error accessing file for config file:///etc/yum.conf   This is what my yum.conf looks like:  # This is the fedorafaq.org yum.conf for Fedora Core 5. # # Note that you also need the rest of the configuration # as described at http://www.fedorafaq.org/#yumconf  [main] cachedir=/var/cache/yum keepcache=0 debuglevel=2 logfile=/var/log/yum.log exactarch=1 obsoletes=1 gpgcheck=1 plugins=1 metadata_expire=1800  # PUT YOUR REPOS HERE OR IN separate files named file.repo # in /etc/yum.repos.d   This is what my yum.repos.d looks like:  fedora.repo fedora-updates.repo fedora-updates-testing.repo   The permissions on those files are 0666 and it won't allow me to make them root:root How to sum up values of each two rows across their line in linux What are the differences between a pointer variable and a reference variable in C++? I know references are syntactic sugar, so code is easier to read and write. But what is the difference between a pointer variable and a reference variable? Can't disable windows 10 PIN complexity I recently linked my Microsoft Office account to my school to have access to Office for students. Right after linking my account, I couldn't login to my computer again with my 4-digit pin code and got the message: "your organization has set the following pin code requirements. Your pincode has to be at least 6 digits long."   I had to change my pincode but I want to change it back to my original 4-digit one. Some internet research has made me enable gpedit.msc, but when I try to find the PIN complexity setting in gpedit (Computer Configuration\Administrative Templates\System\PIN Complexity) it just isnt't there.   Another possible path in gpedit I found on the internet to do this was: Computer Configuration > Administrative Templates > Windows Components > Windows Hello for Business > PIN Complexity, but that also didn't work as I don't have "Windows Hello for Business"   I also tried resetting my computer to the date before I joined my 'organization', but that didn't work either.   Does anyone have an idea how I can resolve this issue? "wait more than" vs. "wait FOR more than" If you make him wait (for) more than ten minutes, he'll just walk away.  I found both instances on Google Books:  and . So I'm a bid confused. What are the differences between 32-bit and 64-bit systems?  If you have used both of them, what kind of sharp differences have you experienced?   Would it be a problem to use 32-bit programs on 64-bit systems in some cases? I have two tables A &amp; B. Table A id | name | num  Table B id | date | roll  id in table B is a foreign key which relates to the id in table A. I want to get the contents of table B where id is given and also along with that data, I need the name whose id is given from table A. NOTE: cannot use multiple queries separated by commas; need to do both the SELECTs in one query. How do I create a vertical version of \not? Getting a FILE* from a std::fstream Is there a (cross-platform) way to get a C FILE* handle from a C++ std::fstream ?  The reason I ask is because my C++ library accepts fstreams and in one particular function I'd like to use a C library that accepts a FILE*. Take into account all of the reputations from all stackexchange sites How to hide the [@] in a Minecraft command block Dihedral group - elements of order $2$ Using Windows 10, I can't run certain commands from PowerShell but they all work correctly using cmd. PowerShell: node --version: node : The term 'node' is not recognized as the name of a cmdlet, function, script file, or operable program. Check the spelling of the name, or if a path was included, verify that the path is correct and try again. At line:1 char:1 + node --version + ~~~~     + CategoryInfo          : ObjectNotFound: (node:String) [], CommandNotFoundException     + FullyQualifiedErrorId : CommandNotFoundException  node.exe --version: no output. node.exe: opens a new black console (node.exe application itself). Get-Host | Select-Object Version: Version ------- 5.1.19041.906  Get-Command node: The term 'node' is not recognized as the name of a cmdlet, function, script file, or operable program. Check the spelling of the name, or if a path was included, verify that the path is correct and try again. At line:1 char:1 + Get-Command node + ~~~~~~~~~~~~~~~~ + CategoryInfo : ObjectNotFound: (node:String) [Get-Command], CommandNotFoundException + FullyQualifiedErrorId : CommandNotFoundException,Microsoft.PowerShell.Commands.GetCommandCommand Get-Command node.exe: CommandType Name Version Source ----------- ---- ------- ------ Application node.exe 14.16.1.0 C:\Program Files\nodejs\node.exe $env:Path output has nodejs &amp; git &amp; other paths. cmd: everything works fine with any command. I have git and nodejs in my PATH env variable. What's wrong with PowerShell, what a strange behavior is this and how to fix it?