1
vote
2answers
45 views

Proving that the line integral $\int_{\gamma_{2}} e^{ix^2}\:\mathrm{d}x$ tends to zero

Let $f(z) = e^{iz^2}$ and $\gamma_2 = \{ z : z = Re^{i\theta}, 0 \leq \theta \leq \frac{\pi}{4} \} $. All the sources I have found online, says that the line integral $$ \left| \int_{\gamma_2} ...
1
vote
0answers
35 views

What are some good general estimates?

For example, the triangle inequality for complex numbers and summations is a good one. Also, the ML-Estimate (Estimation Lemma), Cauchy Estimates $|zw|=|z||w|$. As you can probably notice, I really ...
1
vote
1answer
18 views

Supremum of (e^(i z t) - 1)/z

i'm new here, so i'm not sure if this is the right place to ask this question: I know that the following holds true: $$ \forall\, t \in \mathbb{R} \; \forall\,x\in\mathbb{R}\setminus\{0\} ...
7
votes
1answer
104 views

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 ...
0
votes
0answers
26 views

Question about picking value large enough so that an inequality holds for all values larger than said value

This question makes me wonder about more general inequalities, but I have a particular example. Let $C$ be a positive fixed constant, $0<\epsilon<1$ be given, and assume $\alpha,\beta\in ...
0
votes
1answer
29 views

About complex sum

Let $\left(c_{n}\right)_{n},\,\left(d_{n}\right)_{n}$ two successions of complex numbers and let $N$ a large natural number.Is it true that ...
0
votes
1answer
26 views

inequality, complex number [closed]

let $b>0$, $y:[0,b]\rightarrow\mathbb{C}$ and $y(0)=0$, where $\mathbb{C}$ denotes the space of complex numbers. Is the following inequality true or not? $$|y(x)|\le \max_{\tau\in[0,b]} |y'(\tau)| ...
4
votes
0answers
41 views

Soft Question: Inequalities like this

I am studying signed and complex measure and at a point in a proof the following lemma is being used: Lemma. If $z_1,...,z_n$ are complex numbers, then there exists a subset $S\subset\{1,2,...,n\}$ ...
13
votes
2answers
154 views

Maximum of $|(z-a_1)\cdots(z-a_n)|$ on the unit circle

Let $a_1,\ldots,a_n$ be points on the unit circle. Let $P(z)=(z-a_1)\cdots(z-a_n)$. The maximum principle or Rouche's theorem can be used to show that there exists a point $b$ on the unit circle such ...
0
votes
0answers
119 views

complex sum-integration

Let $a_n \in \mathbb{R}$. Show that, $$0\le \sum_0^N\sum_0^N\frac{a_na_m}{n+m+\frac{1}{2}}\le \pi \sum_0^Na_n^2.$$ Hint: Integrate on the semicircle with radius R, the sum ...
2
votes
0answers
69 views

What is $iav-\log(v)$? Any series expansion or inequality for it?

I am investigating the integral of this question here where \begin{equation} \frac{\exp(i a v)}v=\frac{\exp(i a v)}{\exp(\log(v))}=\exp(iav-log(v)) \end{equation} where I am interested in the ...
3
votes
1answer
62 views

Find the complex $z$ such $\max{(|1+z|,|1+z^2|)}$ is minimum

find the complex $z$,such $$\max{(|1+z|,|1+z^2|)}$$ is minimum My try: let $z=a+bi$,then $$|1+z|=\sqrt{(a+1)^2+b^2}$$ $$|1+z^2|=|1+a^2+2abi-b^2|=\sqrt{(1+a^2-b^2)^2+4a^2b^2}$$ Then I can't,Thank ...
0
votes
1answer
54 views

How to rigorously show that $\frac{z^4}{1+z^6}$ behaves like $\frac{1}{z^2}$ for large $|z|$?

I want to make the integral $$\int_{C_R}\frac{z^4}{1+z^6}dz$$ disappear, but I just realised I'm having trouble writing down a coherent set of inequalities that would allow me to bound it by ...
0
votes
1answer
54 views

Showing an inequality

I wish to show $$|{(Re^{i \theta})^{-\frac{1}{2}}}\exp(\frac{-1}{Re^{i \theta}})| < \frac{M}{R^k}$$ for some M, k > 0 I've managed to reduce it to $$|R^{-\frac{1}{2}}| |\exp(\frac{-1}{Re^{i ...
3
votes
2answers
60 views

The ratio $\frac{u(z_2)}{u(z_1)}$ for positive harmonic functions is uniformly bounded on compact sets

I want to prove the following: If $E$ is a compact set in a region $\Omega \subset \mathbb C$, prove that there exists a constant $M$, depending only on $E$ and $\Omega$, such that every positive ...
1
vote
1answer
59 views

Upper bound on a sum of complex numbers

Let $A=\{z_1, z_2, z_3,\ldots \} $ be a set of complex numbers with $|z_j|\ge 1$ such that the number of elements of $A$ with modulus $<r$, denoted $N_A(r)$, satisfies $$ N_A(r) \le C_0r^N $$ for ...
0
votes
0answers
25 views

Prove that $\int_{{B(0,\epsilon)}\setminus \{z_1=0\}}\det\left(\text{Hessian}_u(z)\right)\mathrm{d}V=\infty$

I have a problem: For $u(z_1,z_2)=\left (-\log\left | z_1 \right | \right )^\alpha\cdot \left ( \left | z_2 \right |^2-1 \right )$, where $\alpha \in \left (0,1 \right )$. Prove that if $\alpha ...
0
votes
3answers
174 views

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: ...
0
votes
1answer
112 views

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 $$\left| \sum_{i=1}^N a_i b_i\right|^2 \le \sum_{i=1}^N |a_i|^2 \sum_{i=1}^N |b_i|^2.$$ ...
0
votes
0answers
26 views

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 ...
1
vote
1answer
39 views

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 ...
2
votes
1answer
102 views

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 $|z_1+z_2|\geq ||z_1|-|z_2||$ ...
3
votes
1answer
63 views

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 ...
1
vote
1answer
61 views

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. ...
2
votes
1answer
72 views

Proving an inequality involving the logarithm

In Ahlfors' complex analysis text he claims that: If $|u| < 1$ we have by power-series development $$ \log \left\lvert E_h(u) \right\rvert \leq \frac{1}{h+1} ...
4
votes
1answer
109 views

An inequality involving arctan of complex argument

I have the following conjecture: \begin{equation} \text{Re}\left[(1+\text{i}y)\arctan\left(\frac{t}{1+\text{i}y}\right)\right] \ge \arctan(t), \qquad \forall y,t\ge0. \end{equation} Which seems to be ...
2
votes
1answer
53 views

An inequality on holomorphic functions

Let $D := \{z \in \mathbb{C}: |z| < 1\}$ and $f\colon D \rightarrow \mathbb{C}$ be holomorphic. Suppose $\lvert f(z)\rvert \leq 1$ on $D$, show that $$\frac{|f(0)| - |z|}{1 + |f(0)||z|} \leq |f(z)| ...
2
votes
2answers
72 views

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 ...
2
votes
1answer
76 views

Inequality holding for complex numbers in the unit disk

In Nehari's book Conformal Mapping he gives it as an exercise to prove that for $a,b\in \mathbb{C}$, $|a|, |b| <1$ we have $$\frac{|a|-|b|}{1-|ab|} \leq \left|\frac{a-b}{1-\overline{a}b}\right| ...
1
vote
1answer
73 views

upper bound on product of distances from points on a circle

Let $C$ be a circle of radius $1$ in the complex plane with $n$ points on the boundary. Provide an upper bound on the product of the distances of a given point on the circle to the other $n$ points. ...
3
votes
2answers
173 views

Application of Rouche's theorem gives two different answers?

So I am supposed to find how many solutions the equation $z^7-5z^4+iz^2-2 = 0$ has in the region $|z|<1$. Here's the dilemma: $|z^7-5z^4+iz^2|= |(-1)(-z^7+5z^4-iz^2)| = |-z^7+5z^4-iz^2| \geq ...
2
votes
0answers
69 views

How to obtain the infimum of this inequalities?

Let $A$ be the family of functions $f(z)=z+a_2z^2+\cdots$ that are analytic in unit disk $D:\{z:|z|<1\}$ and $S$ is the subfamily of functions that are univalent in $D$. $R(a)$ is the subfamily of ...
0
votes
2answers
55 views

Complex number question

For any complex numbers $z_1, z_2$, is the quantity $S$: $$ S = 4 \left(| z_1|^6 + |z_2 |^6\right ) + 4 |z_1|^3 |z_2 |^3 + \left(2 |z_1|^2\times \overline{z_1}^2\times z_2^2\right) + \left(2 ...
3
votes
2answers
67 views

How to prove that there exists a $z_0 \in U_{1} [0]$ such $ \prod_{k=1}^{n} |z_0 - a_k | \geq 1 $ for $a_1, \dots , a_n \in U_{1} [0] $?

Let $a_1 , \dots , a_n $ be points in the unit circle/ball in $\mathbb{C}$ around $(0,0)$ (also known as $U_{1} [0]$), which do not necessarily differ from one another. How to prove that there exists ...
0
votes
2answers
51 views

Some inequality with complex variables and a concavity of a complex function.

I am doing some project. I have to calculate the estimates of an operator. But I was stuck on a part. I need to show the following form of inequality to derive a conclusion what I want to show. ...
2
votes
1answer
103 views

A problem involving Schwarz lemma (from Gamelin)

I have a problem that I cannot solve; it was homework this past semester, I didn't get it then, and now I'm going over past problems and am stuck on it again. It reads-- suppose $f(z)$ is analytic ...
0
votes
0answers
24 views

About an inequality of complex numbers [duplicate]

For $\alpha = 0,1,2,3$, does this inequality always hold for any complex number $z_1, z_2$? $$ | \;\overline{z_1}^{3-\alpha} z_1^\alpha - \overline{z_2}^{3-\alpha} z_2 ^\alpha | \leq C ( |z_1 - z_2 ...
1
vote
0answers
39 views

$ | \;\overline{z_1}^{3-\alpha} z_1^\alpha - \overline{z_2}^{3-\alpha} z_2 ^\alpha | \leq C ( |z_1 - z_2 |^2 + |z_2| ^2 )|z_1 - z_2|$?

I questioned about this inequality before, but how about weaker one: For $\alpha = 0,1,2,3$, does this inequality always hold for any complex number $z_1, z_2$? $$ | \;\overline{z_1}^{3-\alpha} ...
0
votes
1answer
33 views

$ | \;\overline{z_1}^{3-\alpha} z_1^\alpha - \overline{z_2}^{3-\alpha} z_2 ^\alpha | \leq ( |z_1 - z_2 |^2 + |z_2| ^2 )|z_1 - z_2|$?

For $\alpha = 0,1,2,3$, does this inequality always hold for any complex number $z_1, z_2$? $$ | \;\overline{z_1}^{3-\alpha} z_1^\alpha - \overline{z_2}^{3-\alpha} z_2 ^\alpha | \leq ( |z_1 - z_2 |^2 ...
2
votes
1answer
127 views

how to find bounds on (complex) coefficients from bounds on a polynomial?

I'm trying to prove the following two statements about a polynomial $p$ of degree $n$ with complex coefficients: If $|p(x)|\le1$ for all real $x$ with $|x|\le1$, then every coefficient of $p$ has ...
1
vote
1answer
37 views

Complex number inequality calculations

Let $z_1, z_2 \in \mathbb C$. I hope to show that the quantity $$ 3 | z_1 |^4 + 3 |z_2|^4 + 4 |z_1 |^2 |z_2 |^2 + z_2^2 \overline{z_1} ^2 + z_1^2 \overline{z_2}^2 $$ is nonnegative. Does this hold? I ...
0
votes
1answer
88 views

Show that $|\sin(z)|≥1$ at all points on the square with vertices $±(N+1/2)π±(N+1/2)πi $, for any positive integer $ N $.

Show that $|\sin(z)|≥1$ at all points on the square with vertices $±(N+1/2)π±(N+1/2)πi$, for any positive integer $N$. One of the confusing things is 'at all points on the square with ...'. I tried ...
3
votes
1answer
127 views

Prove the inequality?

Let $f$ be an analytic function in the unit disc without zeros satisfying $|f|\leqq 1$. Prove that $$ \sup_{|z\leqq{1/5}|}|f(z)|^2\leqq \inf_{|z|\leqq{1/7}}|f(z)| $$ Help me please. These questions ...
7
votes
1answer
126 views

Prove $|Im(z)|\le |\cos (z)|$ for $z\in \mathbb{C}$.

Let $z\in \mathbb{C}$ i.e. $z=x+iy$. Show that $|Im(z)|\le |\cos (z)|$. My hand wavy hint was to consider $\cos (z)=\cos (x+iy)=\cos (x)\cosh (y)+i\sin (x)\sinh(y)$ then do "stuff". Then I have ...
3
votes
1answer
57 views

Complex integral Q

I have this question: Let C be an open (upper) semicircle of radius R with its centre at the origin, and consider $\int_C f(z) \, dz$ where $f(z)=\frac 1{z^2 + a^2}$ for real $a > 0$. Show that ...
2
votes
1answer
93 views

A Growth Inequality on $\mathbb{C}$-Polynomials

For which class of polynomials over $\mathbb{C}^{n}$ does the following growth inequality hold? For any multi-index $\alpha$, there are positive constants $A, B, C, D < \infty$ such that ...
4
votes
1answer
328 views

A qualifying exam problem involving Schwarz lemma

This is a problem in the book "Berkeley Problems in Mathematics", which I think the solution given is wrong, can someone help? The following problem appeared in Spring 1991. Let the function $f$ ...
2
votes
3answers
159 views

Prove inequality in complex numbers in an unit circle

Given $|\omega| < 1$, $\omega \neq 0$ and $|z| < 1$. Prove inequality: $$\frac{|\frac{|\omega|}{\omega}z+1|}{|1-z \bar \omega|} \le \frac{2}{1-|z|}$$ It is simple but i have problems with it. ...
2
votes
1answer
61 views

Prove, that $f: S_{k} \rightarrow \mathbb{C}-\{0\}$ is a surjection.

For $k>0$ define $S_{k} := \{z=x+iy\in\mathbb{C}\mid |z|<k,\ \ \ \ k\cdot y>|x|\}\subset\mathbb{C}$ Let $f(z)=\exp(1/z)\ \ \ \text{for}\ \ \ z\neq 0$ Prove, that $f: S_{k} \rightarrow ...
2
votes
2answers
83 views

Complex function inequality

Let $\lambda \in (0, 1)$ . Suppose, that function $f : \mathbb{C} \rightarrow \mathbb{C}$ satisfy the inequality $|f(u) - f(v)| \leq \lambda|u-v|$ Prove, that for all $a \in \mathbb{C}$ $z = f(z) + ...