# Tagged Questions

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### Show that $|z+1|\le|z+1|^2 +|z|$ for all $z \in \mathbb{C}$

Question: Show that $|z+1|\le|z+1|^2 +|z|$ for all $z \in \mathbb{C}$ So far I have, Suppose $1\le|z+1|$ $|z+1|\le|z+1|^2$ $|z+1|\le|z+1|^2+|z|$ Now I must show $|z+1|<1$ but this is where ...
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### Prove that for holomorphic function is inequality $M|a_1| \le M^2 - |a_0|^2$

Let $$f(z) = \sum_{k=0}^{\infty}a_kz^k$$ be holomorphic function in unit disc and $f(z) < M$ for $|z|<1$. Show that $$M|a_1| \le M^2 - |a_0|^2$$ I have any ideah how can I prove this ...
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### About B. Ya Levin's proof that $|f(x)| \leq M$ implies $|f(x+iy)| \leq Me^{\sigma y}$

This question is about Theorems 1 through 3 on pages 37-38 of B. Ya Levin's Lectures on Entire Functions, available on Google Books. If you can't access the Google Books link there is also a ...
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### How find this maximum of this complex numbers of $x,y$

let $x,y$ be complex numbers,such that $|x|=|y|=1$. Can anyone help me to find the maximum value of the following expression $$|1+x|+|1+xy|+|1+xy^2|+\cdots+|1+xy^{2013}|-1007|1+y|$$ my try: ...
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### Cauchy's inequality

Let $\{a_i\}_{i=1}^N$ and $\{b_i\}_{i=1}^N$ be two sets of complex numbers. Prove Cauchy's inequality $$\left| \sum_{i=1}^N a_i b_i\right|^2 \le \sum_{i=1}^N |a_i|^2 \sum_{i=1}^N |b_i|^2.$$ ...
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### Questions about $|f(1+a+bi)|<|f(1+a)|$

Let $a,b >0$ and $|*|$ denote the absolute value. Let $f(z)$ be a realvalued analytic function defined for $Re(z)>1.$ For any $a,b$ we have $|f(1+a+bi)|<|f(1+a)|$. Some questions : $1)$ If ...
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### Bounding the function $(-z)^{s-1}$ over the square with vertices $(\pm(2n+1) \pi,\pm(2n+1) \pi)$

In Ahlfors' Complex analysis text, page 216 he claims that $\left \lvert (-z)^{s-1} \right \rvert$ is bounded by a multiple of $n^{\sigma+1}$ over the square contour $C_n'$ with vertices in ...
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### Regarding the derivation of triangle inequality related inequality (undergraduate complex analysis)

I am using Brown and Churchill's Complex Analysis Textbook, and on pg.11 of the eighth edition, there is a triangle inequality derivation as followed to prove $|z_1+z_2|\geq ||z_1|-|z_2||$ ...
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### Which statement of Hadamard's factorization theorem is true?

In this wikipedia article it says that if the order $\rho$, and the genus $g$ of an entire function can satisfy the equation $$g=\rho+1,$$ if the order is an integer. However, in Ahlfors' Complex ...
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### Complex number inequality, $|e^{z_1}-e^{z_2}| \leq |z_1 - z_2|$ if $Re(z_1),Re(z_2) \leq 0$

I'm trying to show the complex inequality $|e^{z_1}-e^{z_2}| \leq |z_1 - z_2|$ holds if $Re(z_1),Re(z_2) \leq 0$. It seems intuitively obvious but I haven't been able to find something that works. ...
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### For analytic $f$ on $D_2(0)$ with $|f(z)| \le |\sin z|$ on $\partial D_2(0)$ , show $|f(\frac{\pi}{2})| \le \frac{4}{\pi}$
Let $f$ be analytic on $D_2(0)$ and continuous up to the boundary with $|f(z)| \le |\sin z|$ on $\partial D_2(0)$. Prove that $|f(\frac{\pi}{2})| \le \frac{4}{\pi}$. This problem appears on an old ...