Linked Questions

10
votes
0answers
924 views

Enestrom-Kakeya Theorem [duplicate]

The Enestrom-Kakeya theorem states that all roots of the polynomial: $$p(z):=\sum_{k=0}^n a_kz^k$$ lie outside the open unit disk if the sequence $(a_k)$ is positive and decreasing. A proof can be ...
0
votes
1answer
258 views

Show that a complex polynomial of degree $n$ doesn't have zeros in a unit ball [duplicate]

Let $n>1$ and $c_0>c_1>c_2>\dots >c_n>0$ and $f(z)=c_0+c_1 z+\dots + c_n z^n$. Show that this polynomial doesnt have zeros in a unit ball $B(0,1)$. Can you give me some feedback?
0
votes
0answers
62 views

Prove all zeros of a polynomial lie in $\{|z| > 1 \}$ [duplicate]

I want to solve following problem. I tried to solve it with "Cauchy integral formula" but I couldn't. How can I solve this?! Suppose $p_0 > p_1 > p_2 > ... >p_n > 0$. Prove that all ...
0
votes
0answers
37 views

Roots of complex polynomial have modulus less than 1 [duplicate]

Let $0<a_0<a_1<...<a_n$, $a_i\in \mathbb R$. I need to show that if $$a_0+a_1z+...+a_nz^n=0$$ then $|z|<1$. Any hints? I don’t know how to begin. I can’t use Rouche’s theorem as it ...
2
votes
0answers
34 views

A problem in functions of a complex variable [duplicate]

Given $n+1$ real numbers $a_0,a_1,\cdots, a_{n}$ such that $0<a_0<\cdots<a_n$. Let $P(z)=\sum\limits_{i=0}^n a_i z^i $, where $z\in \mathbb{C}$. How to show that there must be $n$ roots of $P(...
0
votes
0answers
20 views

Proving that a complex polynomial has no roots in the unit disk [duplicate]

Let $\sum_{k=0}^{n}a_{k}z^{k}$ be a polynomial of degree $n$ with real coefficients satisfying $$a_{0}>a_{1}>....>a_{n-1}>a_{n}>0$$ Prove that $p(z)=0$ implies $\left|z\right|>1$....
5
votes
6answers
401 views

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$. My attempt: As the polynomial is a cubic, it must have atleast one real ...
6
votes
3answers
343 views

Proving that $\sum\limits_{n = 0}^{2013} a_n z^n \neq 0$ if $a_0 > a_1 > \dots > a_{2013} > 0$ and $|z| \leq 1$

I'm going to teach a preparation course for the complex analysis qualifying exam from my university (which basically consists of me doing some problems from past exams) and I'm trying to solve some ...
4
votes
1answer
123 views

Let $a_n$ be a decreasing sequence. Prove that the power series $\sum a_n x^n$ has no roots in $A=\{z\in C:|z|<1\}$

Let $a_n$ be a decreasing, positive sequence, in the real space. Prove that the power series $\large p(z)=\sum\limits_{k=0}^{n} a_k z^k$ has no roots in $A=\{z\in C:|z|<1\}$. What I did so far $zp(...
1
vote
1answer
228 views

Prove the equation has no root in the circle $|z| < 1$

Suppose that $0 < a_0 \le a_1 \le \dots \le a_n$. Prove that the equation $$P(z) = a_0z^n + a_1z^{n-1} + \dots + a_{n-1}z + a_n = 0$$ has no root in the circle $|z| < 1$.
4
votes
1answer
89 views

On a mistake in a derivation regarding a recurrence relation.

Suppose we have a sequence $\{u_t \}_{t \in \mathbb{N}}$ given by the recurrence relation $$u_{t+1} = q_0 u_t + q_1 u_{t-1} + \dots + q_p u_{t-p} + \epsilon, \quad \epsilon >0$$ where $p \in \...
1
vote
1answer
95 views

Proving that a complex polynomial has no roots on the unit circle

Let $\sum_{k=0}^{n}a_{k}z^{k}$ be a polynomial of degree $n$ with real coefficients satisfying $$a_{0}>a_{1}>....>a_{n-1}>a_{n}>0$$ Prove that $p(z)=0$ implies $\left|z\right|>1$....
1
vote
1answer
120 views

How to prove $a_0 + a_1 \cos \theta + a_2 \cos 2\theta + \cdots + a_n \cos n \theta$ has $2n$ different zeros, $\theta \in (0,2\pi)$.

$0 < a_0 < a_1 < \cdots < a_n$. Prove that $a_0 + a_1 \cos \theta + a_2 \cos 2\theta + \cdots + a_n \cos n \theta$ has $2n$ different zeros, $\theta \in (0,2\pi)$. [Hint: First prove that $...
1
vote
0answers
136 views

Polynomial $p(z)$ in $\mathbb{C}$ has no zero whose modulus does not exceed $1$ [duplicate]

For $n>1$ consider real numbers $c_0>c_1>.....>c_n>0$. Prove that the polynomial $$p(z) = c_0+c_1z+.....+c_nz^n$$ in $\mathbb{C}$ has no zero whose modulus does not exceed $1$.
2
votes
0answers
99 views

A question about roots of a complex polynomial [duplicate]

Possible Duplicate: Enestrom-Kakeya Theorem Suppose that $0<a_0\leq a_1\leq\cdots\leq a_n$, then prove that the complex polynomial $$P_n(z)=a_0z^n+a_1z^{n-1}+\cdots+a_{n-1}z+a_n$$ cannot have ...