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 ...
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?
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 ...
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 ...
34 views

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 $|z| < 1$.
89 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 $1$.
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 ...