0
votes
0answers
25 views

Proof of inequality related to complex function

Let $f(z)$ be a nonsingular complex function whose domain is $D=\{z\in C\ ;\ |z|<R\}$. $f(z)$ satisfies $f(0)=1$, and Re$f(z)$ is positive everywhere on $D$. Let $g(z)$ be $$g(z) = ...
-1
votes
2answers
42 views

Upper bound on the rational function of $z$ in terms of $|z|$

Show that: $$\frac{|2z^2-5|}{|z^2+1||z^2+4|} \le \frac{2|z|^2+5}{(|z|^2-1)(|z|^2-4)}$$ I started by considering that for the above to hold $|2z^2-5|\le (2|z|^2+5)$, and $|z^2+1|\ge (|z|^2-1)$, ...
2
votes
1answer
17 views

Estimate of line integral of O(x^n) function

Let $f$ be an analytic function in some sector in the complex plane behaving as $$f(z)=\mathcal O(z^n)$$ for some $n$ as $z\to\infty$. Can one prove in general that line integrals of $f$ (in this ...
2
votes
1answer
30 views

If $w_1=a_1+ib_1$ and $w_2=a_2+ib_2$ are complex numbers, then $|e^{w_1}-e^{w_2}|\geq e^{a_1}-e^{a_2}$

Let $w_1=a_1+ib_1$ and $w_2=a_2+ib_2$ be two complex numbers. Ahlfors says that $|e^{w_1}-e^{w_2}|\geq e^{a_1}-e^{a_2}$. I don't understand why that is. Any help would be greatly appreciated.
2
votes
1answer
31 views

Inequations on holomorphic functions

Let $f : \mathbb{C}\setminus\{0\} \to \mathbb{C}$ an holomorphic function such as $$\exists C_0 > 0 / \forall z \in \mathbb{C}\setminus\{0\} \qquad |f(z)| \leqslant C_0\left( |z| + ...
3
votes
4answers
164 views

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 ...
0
votes
0answers
20 views

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 ...
0
votes
2answers
15 views

$|z_{1}- z_{2}| \leq |w_{1}- w_{2}| \implies |c_{1}z_{1}- c_{2}z_{2}| \leq |c_{1}w_{1}- c_{2} w_{2}|$?

Let $z_{1}, z_{2}, w_{1}, w_{2} \in \mathbb C$ with $|z_{1}- z_{2}| \leq |w_{1}- w_{2}|.$ Fix $c_{1}, c_{2}\in (0, \infty).$ My Question is: Can we expect, $|c_{1}z_{1}- c_{2}z_{2}| \leq ...
0
votes
1answer
26 views

Bound Function / Exponential (Unit Circle)

I need to mathematically prove the next inequality: $$\frac{(e^{2x}-1)^2}{(e^{2x}+1)^2} \le 1$$ If I graph the function, it is bounded above by 1, I don't know however how to proceed with this ...
8
votes
1answer
163 views

How find the maximum value of $|bc|$

Question: Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$ It is said this is answer is ...
1
vote
1answer
24 views

A proof of a Proposition by Hurwitz

Here is a proof of a Theorem by Hurwitz. (Source: G. De Marco, Selected topic in Complex Analysis.) (In these notes, $B(c,r]$ means the closed ball in $\mathbb{C}$ centered in $c\in\mathbb{C}$ and ...
1
vote
2answers
89 views

Intuition or figure for Reverse Triangle Inequality $||\mathbf{a}| − |\mathbf{b}|| ≤ |\mathbf{a} − \mathbf{b}|$ (Abbott p 11 q1.2.5)

I acquiesce to Wikipedia's picture for Triangle Inequality. But without referring to Triangle Inequality at all, is there intuition or figure please for Reverse Triangle Inequality for all ...
1
vote
1answer
30 views

Plotting a complex inequality

So I am looking at the exercise: "Plot $|z-3i| + |z-4| > 7 $ in the complex plane." I have done similar exercises by using $z = x + iy $ and treating the problem as a real valued inequality, but ...
1
vote
3answers
49 views

Range of modulus of Complex Number

If $z\in \mathbb{C}$ and $$ |z-1|+|z+3|\le 8$$ Find the Range of $$|z-4|$$ My Try: $$|2z-8|=|2z+2-10|\le |2z+2|+10=|(z-1)+(z+3)|+10\le|z-1|+|z+3|+10\le18$$ $\implies$ $$|z-4|\le9$$ I need Hint to ...
0
votes
0answers
9 views

Proving an inequality on $G_n(z)=a_0^n G_0(z)+n a_1^{n-1} G_1(z)+\frac{n (n-1)}{2} a_2^{n-2} G_2(z)+…$

Hypothesis: 1) $a_{n,r}, a_{n,i}\in \mathbb R$ such that $$a_n=a_{n,r}+i a_{n,i}\in \mathbb C$$ and $$|a_{n,r}|\leq k<1,\ \ \ |a_{n,i}|\leq h<1$$ 2) $|z|\leq R, \ \ |G_m(z)|\leq M m!$ ...
1
vote
0answers
106 views

Normal sequences and Montel's Theorem

I am currently stuck on an exam question involving normal sequences and Montel's theorem: Give two examples of non-constant normal sequences one in the $(a)$ unit disk $\mathbb{D}$ and one in $(b)$ ...
0
votes
1answer
28 views

$C^{-1} (1+|x|^{2})^{\frac{s}{2}} \leq (1+|x|)^{\frac{s}{2}} \leq C (1+|x|^{2})^{\frac{s}{2}}$?

Let $s\in \mathbb R,$ and define $f: \mathbb R^{n}\to [0, \infty)$ such that $f(x)= (1+|x|^{2})^{\frac{s}{2}}, (x\in \mathbb R^{n})$ and $g:\mathbb R^{n}\to [0, \infty)$ such that $g(x)= ...
1
vote
2answers
70 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
39 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
20 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
126 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
31 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
31 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
29 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
49 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
162 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
124 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
64 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
77 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
61 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
27 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
178 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
126 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
43 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
108 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
69 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
64 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
77 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
114 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
58 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
74 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
84 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
81 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
198 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
76 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
57 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 ...