Tagged Questions

0
votes
2answers
50 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
52 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
29 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
48 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
21 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
30 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
27 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
72 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
30 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
44 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
69 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
73 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
41 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
90 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
175 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
106 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
60 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
70 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) + ...
3
votes
1answer
242 views

Application of Schwarz lemma

Each analytic function mapping the right half complex plane into itself must satisfy $$ \left|\frac{f(z)-f(1)}{f(z)+f(1)}\right| \leqslant \frac {z-1}{z+1}$$ for $\text{Re}\; z > 0.$ I have a ...
0
votes
1answer
36 views

Integration of a function in Pommerenke's Univalent Function

I trying to understand a proof in Pommerenke's Univalent Functions. Given $f$ analytic on the unit disk and $M(r) := \max\limits_{0 \leq \theta \leq 2\pi}|f(re^{i\theta})|$, Using the fact that ...
8
votes
1answer
303 views

Proof using Cauchy–Schwarz Inequality.

This is a problem I am trying to solve for a couple of months without any success. I found it in a paper and according to the authors can be proved using Cauchy–Schwarz inequality. Let $f(x)$ be a ...
2
votes
3answers
118 views

Contour integral in complex analysis

Show that if C is the boundary of the triangle with vertices at the points $0,3i$ and $-4$ oriented in the counterclockwise direction, then $$\bigg|\displaystyle \int_C(e^z-\overline{z})dz\bigg| \leq ...
0
votes
0answers
146 views

Equality in the Cauchy-Bunyakowsky-Schwarz inequality for a semi-inner product

I am stuck with an exercise that I found in a textbook by Conway. First, I would like to clarify what is meant by a semi-inner product. Definition. Suppose that $\mathscr X$ is a vector space over ...
4
votes
3answers
403 views

Proving two integral inequalities

Can anyone help me to prove that these integral inequalities hold? Here $x$ is a real value: $$ \left| \int_a^b\ f(x) dx \right| \leq \int_a^b\ |f(x)| dx $$ Here $z$ is a complex value: $$ \left| ...
4
votes
3answers
140 views

Prove this inequality for $|z| \leq 1$

I am supposed to prove that $$\frac{|e^z-1|}{e-1} \leq |z|$$ for $|z| \leq 1$. My guess is that I have to show that the LHS $\leq 1$ and then apply Schwarz's Lemma. But I am not able to prove that!
0
votes
1answer
54 views

A (not for me) simple inequality

I have some problems to understand the following situation: Let $A=\{z\in\mathbb C : |z|\ge 1$ and $|\Re(z)|\le\frac{1}{2}\}$, then the inequality $|cz+d|\le 1$ with $c,d\in\mathbb Z$ doesn't have ...
5
votes
3answers
403 views

Derivatives of the Riemann zeta function at $s=0$

It's a curious fact that for $n>0$, $\zeta^{(n)}(0)\approx -n!$. Apostol gave a table for $\frac{\zeta^{(n)}(0)}{n!}$, among other results on $\zeta^{(n)}(0)$ . the sequence : $$\delta_{n}=\left | ...
0
votes
3answers
145 views

Complex inequality

How can I show this inequality $\sqrt{2}|z|\geq |\mathrm{Re} (z)|+|\mathrm{Im}(z)| $ please give me some hint. Which result is useful to show this. please help me out.thanks in advance.
8
votes
2answers
246 views

How to prove a property regarding periodicities of points in the Mandelbrot set?

While studying a visual representation the Mandelbrot set, I have come across a very interesting property: For any point inside the same primary bulb (a circular-like 'decoration' attached to the ...
3
votes
2answers
96 views

Norm inequality on Complex Numbers.

For $z,w \in \mathbb{C}$, it is true that $ 2 | z w| \leq |z|^2 + |w|^2 $. How does this imply the identity: $$|z+w|^2 \leq 2(|z|^2 + |w|^2 )? $$
1
vote
3answers
602 views

Triangle Inequality Proof

I need this one result to do a problem correctly. I want to show that for any $b \in \mathbb{C}$ and $z$ a complex variable: $$ |z^2 + b^2| \geq |z|^{2} - |b|^{2}$$ My attempts have only led me to ...
1
vote
1answer
351 views

Equality in the Schwarz-Pick theorem implies function is a linear fractional?

Part of the Schwarz-Pick Theorem states that for an analytic automorphism of the unit disk, then $$ \frac{|f'(z)|}{1+|f(z)|^2}\leq\frac{1}{1-|z|^2}. $$ In the wikipedia article of the ...
4
votes
1answer
119 views

Why does $\frac{|f(z)-f(z_0)|}{|f(z)-\overline{f(z_0)}|}\leq\frac{|z-z_0|}{|z-\bar{z}_0|}$ when $\mathrm{Im}z>0\implies\mathrm{Im}f(z)\geq 0$?

I'm trying to understand the following inequality. Let $f$ be holomorphic, such that $\mathrm{Im}f(z)\geq 0$ when $\mathrm{Im}(z)>0$. Why is it that $$ ...
1
vote
3answers
125 views

Inequality relating diameter of the image of a holomorphic function on the unit disk to the derivative at 0. [duplicate]

Possible Duplicate: First derivative bounded by supremum of difference of values in disc Let $f$ be holomorphic in the disk $D_1(0)$ and let $d=\operatorname{diam}(f(D_1(0))$. I want to ...
2
votes
0answers
115 views

another inequality involving complex numbers.

Let $\{z_i\}$, $i=1,2,\ldots,n$ be a set of complex numbers. Then I know that there is a set $J$ such that $$\left|\sum_{j\in J} z_j\right|\ge \frac{1}{\pi} \sum_{k=1}^n |z_k|. $$ However, how do I ...
1
vote
3answers
98 views

finding bound for the integral

I am trying to get bound for the following integral $$ \int_0^{\infty}\frac{1}{|x|^r}dx, \mbox{for } 1\leq r< \infty $$ In particular, the bound of the form $\frac{constant}{r}$. Sorry, we can ...
4
votes
1answer
182 views

Find number of roots in some area (Rouché's theorem)

The task is to find number of $ {z^4} + {z^3} - 4z + 1 = 0$ in the area $1 < \left| z \right| < 2$. (this task is in the Rouché's theorem paragraph) I used this theorem many times, but I ...
1
vote
0answers
98 views

Bound on Bessel function of the first order

Let $I_1(z)$ be the Bessel function of the first order with purely imaginary argument. Can we explicitly bound $I_1$ on $[0,x]$, where $x>0$ is a real number in terms of $x$?
3
votes
3answers
487 views

Order of growth of $(s-1)\zeta(s)$

Again, order of growth problems. Show that the function $(s-1)\zeta(s)$ is an entire function of growth order $1$; or equivalently, $$|(s-1)\zeta(s)| \leq A_{\epsilon} \; \exp ...
1
vote
0answers
37 views

Lower bounds for holomorphic functions on annuli with explicit bounds on their power series

Let $f$ be a holomorphic function on $\mathbf{C}$ and consider its restriction to the annulus $X=B(0,1) - B(0,3/4)$ in the complex plane. (Here $B(0,r)$ is the open disc with radius $r$ around $0$.) ...
0
votes
1answer
123 views

Inequality question (with complex numbers)

I was wondering how would you derive/get $|q(z)|\ge R^2-|a|R-|b|>R^2/2 $? Thanks.
1
vote
1answer
210 views

How do you prove the following inequality concerning complex Logarithms?

If $0<|w|<1/2$, then $2|w|/3<|\operatorname{Log}(1+w)|$ using power series and modulus inequalities.
2
votes
2answers
130 views

How do I show $\|(e^{-2\pi ihx_j} - 1)/h \| \leq 2 \pi \|x\|$?

How do I show $\|(e^{-2\pi ihx} - 1)/h \| \leq 2 \pi \|x\|$? for each h and x? I thought about using taylor expansion of Euler's formula, but it did not work out. Thank you in advance.
2
votes
1answer
80 views

Bounding a Complex Polynomial

Given the complex polynomial $P(z) = z^2 + a_1z + a_0$ and the constraint that $|z| > 1$, I'm trying to show that $|P(z)| \geq |z|^2 - |a_1||z| - |a_0|$. The obvious thing to do here of course is ...
1
vote
2answers
516 views

A converse theorem to the Cauchy's estimation

This is a follow-up question to About the limit of the coefficient ratio for a power series over complex numbers. Cauchy's estimation in complex analysis is a consequence of the Cauchy's integral ...