All Questions
Tagged with complex-analysis inequality
528
questions
6
votes
0
answers
168
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Show that $(1-z_1\overline{w_1})(1-z_2\overline {w_2})+(1-z_1\overline {w_2}) (1-z_2\overline {w_1})$ is non-vanishing.
Show that $\left (1-z_1\overline{w_1} \right ) \left (1-z_2\overline {w_2} \right )+ \left (1-z_1\overline {w_2} \right ) \left (1-z_2\overline {w_1}\right )$ is non-vanishing (does not take the value ...
- 2,612
1
vote
1
answer
34
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Are the projection functions on the space of functions $\sum_j a_j e^{i \lambda_j x}$ continuous wrt the sup-norm?
In my previous question I was interested in an explicit lower bound for $ \displaystyle \sup_{x \in \mathbb R} \big |\sum_j a_j e^{i \lambda_j x}\big|$ in terms of the coefficients $a_j$. It turns out ...
- 10.4k
1
vote
1
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55
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Area & Perimeter of region $S$ containing points of the form $a+b\omega+c\omega^2$ where $a,b,c \in [0 , 1], \omega=-\frac 12+i\frac{\sqrt 3}{2}$ [closed]
Question: Let $\omega=-\frac 12+i\frac{\sqrt 3}{2}$, and $S$ denotes the set of all the complex numbers in the Argand plane of the form $\boldsymbol{a+b\omega+c\omega^2}$, where $a,b,c \in [0 \, , \, ...
- 81
4
votes
2
answers
138
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Let $a,b,c,d\in\mathbb{C}$ such that $|a|+|b|\leq 1$ and $|c|+|d|\leq 1$. Show that $|3a+b+3c-d|+|a+3b-c+3d|\leq 7$.
Let $a,b,c,d\in\mathbb{C}$ such that $|a|+|b|\leq 1$ and $|c|+|d|\leq 1$. Show that $|3a+b+3c-d|+|a+3b-c+3d|\leq 7$, or find a counterexample -- I don't know for sure that the inequality stated in the ...
- 101
4
votes
3
answers
156
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Is Schwarz's Lemma true for squares?
In a recent complex analysis exam, we were asked which step(s) in the proof of the Riemann Mapping Theorem fail, when you replace every instance of the open unit ball $\mathbb{E}$ with the square
$$\...
- 3,044
0
votes
1
answer
56
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A question of complex numbers involving inequality
$\begin {aligned}|a(z_2-z_3)+(z_3-z_1)|&\ge |a|\, |z_2-z_3|+|z_3-z_1|\text{(by triangle inequality)}\\
&\ge 2\sqrt a\cdot \sqrt{|z_2-z_3|\cdot|z_3-z_1|} \text{(by AM-GM inequality)}
\end{...
- 81
35
votes
4
answers
1k
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Prove that $|z^2+1|\le 2$ implies $|z^3+3z+2|\le 6$
Show that $$\{z \in \mathbb{C}: |z^2+1|\le 2 \} \subseteq \{z \in \mathbb{C} : |z^3+3z+2|\le 6 \} \tag{*}$$
(In other words: Let $z\in \mathbb{C}$ satisfy $|z^2+1|\le 2$. Prove that $|z^3+3z+2|\le 6.$)...
- 3,360
1
vote
0
answers
31
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Constructing such a $0<\delta<1$ that $|z + w|\leq \delta |z|+|w|$ when $0<\varepsilon\leq |\mathrm{Arg}(z)-\mathrm{Arg}(w)|\leq 2\pi -\varepsilon$
Let $z,w\in\mathbb{C}$ be two non-zero complex numbers such that for some $\varepsilon > 0$ we have $$0<\varepsilon\leq |\mathrm{Arg}(z)-\mathrm{Arg}(w)|\leq 2\pi -\varepsilon$$
I am trying to ...
- 1,221
0
votes
1
answer
20
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Trouble proving inequality regarding the modulus of a complex number sum in uniform distribution theorem
I'm currently working through the proof of the theorem stated in Uniform Distribution of Sequences by Harald Niederreiter and L. Kuipers. The theorem states that if $\Delta x_n = x_{n+1} - x_n \...
4
votes
1
answer
80
views
Applying Whittaker's and Watson's Lagrange theorem to $\sin$
I want to use the Lagrange inversion theorem from Whittaker and Watson, p. 133, to find the power series of $\sin^{-1}$ at the origin. It should be a valid procedure because $\sin^{-1}$ is analytic at ...
- 667
2
votes
1
answer
88
views
Find all entire functions such that $|f(z+z')|\leq |f(z)| + |f(z')|$, for all $z,z'\in\mathbb{C}$
Find all entire functions such that $|f(z+z')|\leq |f(z)| + |f(z')|$, for all $z,z'\in\mathbb{C}$
In particular, let $z=z'$ yields $|f(2z)|\leq2|f(z)|$. This gives that $\frac{f(2z)}{f(z)}=c, $ for ...
- 191
2
votes
1
answer
53
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Question about three lines theorem
I'm trying to prove that the function defined as $$F_\epsilon(z)=F(z)M_0^{z-1}M_1^{-z}e^{\epsilon z(z-1)},$$ where $F$ is an holomorphic function on $0<Re z<1$ and continuous and bounded on the ...
0
votes
0
answers
37
views
Convergence of an aggregate remainder in a logarithmic expansion
Let $(x_j)_j$ be an ergodic and stationary sequence of random variables such that $E[x_j]=0$ (for all $j$) and:
$$\frac{1}{n}\sum_{j=0}^{n}f(x_j) \to E[f(x)] < \infty, \quad (n \to \infty)$$
for ...
- 75
7
votes
2
answers
313
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If the roots of $z^4+az^3+bz^2+z$ are distinct and concyclic in the complex plane, does $ab\in\mathbb R$ imply $1<ab<9$?
HMMT February 2022, Team Round, Problem 6 (proposed by Akash Das) is:
Let $\operatorname{\it P\!}{\left(x\right)}=x^4+ax^3+bx^2+x$ be a polynomial with four distinct roots that lie on a circle in the ...
- 401
0
votes
1
answer
79
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Is this inequality for the Gamma function true?
In the book Asymptotics and Mellin-Barnes integrals by Paris and Kaminski (2001), the following inequality is given for the ratio of two gamma functions (Section 2.1.3, page 34) :
For $b-a\ge 1, a\ge ...
