4
votes
1answer
74 views
How prove this polynomials inequality
Let $f=a_0x^n+a_1x^{n-1}+\ldots+a_{n-1}x+a_n\in \mathbb{R}[X]$ a polynomial which has the roots contained in $(-1,1)$. Prove that:
$$\left|\frac{a_1+a_3+a_5+\ldots}{a_0+a_2+a_4+\ldots}\right|<1$$
...
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
36 views
Lower bound for polynomial with complex coefficient
Let $p(z)=z^{n}+a_{n-1}z^{n-1}+...+a_{1}z+a_{0}$ be a polynomial with complex coefficients. Define $R:=1+\sum_{k=0}^{n-1}|a_k|$. Show that $|p(z)| > R$ for all $z \in \mathbb C$ with $|z|>R$.
...
5
votes
2answers
66 views
Half-symmetric, homogeneous inequality
Let $x,y,z$ be three positive numbers. Can anybode prove the follwing inequality :
$(x^2y^2+z^4)^3 \leq (x^3+y^3+z^3)^4$ (or find a counterexample, or find a reference ...)
1
vote
0answers
51 views
How to solve systems of polynomial inequalities?
I am currently working on a project that deals with systems of inequalities and so far I have found algorithms for the basic case of a system of inequalities as well as the non-strict linear ...
4
votes
0answers
63 views
Polynomial bound
Let $P(x)=a_4 x^4+a_3 x^3+a_2 x^2+a_1 x+a_0$ such that
$$\forall i\in \{0, 1, 2, 3, 4\};\phantom{;}a_i\in\mathbb{Z} \wedge |a_i|\leq T\phantom{.}(T\in\mathbb{Z}^+ )$$
Suppose that $P(x)> 0$ for all ...
7
votes
1answer
302 views
+100
Let $a_{i} \in\mathbb{R}$ ($i=1,2,\dots,n$), and $f(x)=\sum_{i=0}^{n}a_{i}x^i$ such that if $|x|\leqslant 1$, then $|f(x)|\leqslant 1$. Prove that:
Let $a_{i} \in\mathbb{R}$ ($i=1,2,\dots,n$), and $f(x)=\sum_{i=0}^{n}a_{i}x^i$ such that if $|x|\leqslant 1$, then $|f(x)|\leqslant 1$. Prove that:
$|a_{n}|+|a_{n-1} | \leqslant 2^{n-1}$.
...
0
votes
2answers
58 views
Minimum value of a polynomial of degree 2.
Let $f:\mathbb R\to \mathbb R$ be a map defined as $f(x)=ax^2+bx+c$, we can rewrite this map as $f(x)=a(x+\frac{b}{2a})^2-\frac{\Delta}{4a}$, where $\Delta=b^2-4ac$. If $a\gt 0$, then the minimum ...
1
vote
1answer
88 views
Inequality involving roots of a third degree polynomial
Let $a,b$ be two positive numbers such that $a^3 \gt 27b$. Consider the polynomial
$$
W(x)=x^3-2ax^2+a^2x-4b
$$
Then we have
$$
W(0)=-4b \lt 0, \ W(\frac{a}{3})=\frac{4}{27}(a^3-27b) \gt 0, \ ...
4
votes
2answers
73 views
$\sum_{i=1}^{n-1} \left|\dfrac{a_ia_{n-i}}{a_n}\right| \geq C_{2n}^n-1$
Given that the equation $$p(x)=a_0x^n+a_1x^{n-1}+\dots+a_{n-1}x+a_n=0$$ has $n$ distinct positive roots, prove that
$$\sum_{i=1}^{n-1} \left|\dfrac{a_ia_{n-i}}{a_n}\right| \geq C_{2n}^n-1$$
I had ...
2
votes
1answer
132 views
Proof of a lower bound of the norm of an arbitrary monic polynomial
In my course I have come across the following problem:
The Chebyshev polynomial of degree $n$, $T_n(x)$, is defined on $[-1,1]$ by $T_n(x)=\cos n\theta$.
Let $q_{n+1}(x)$ be any monic ...
1
vote
2answers
87 views
Three inequalities with sums of fractions over two positive integers
In a proof, I arrive at three inequalities for all $p,q \geqslant 0$:
\begin{align}
\frac{p+1}{q+1} + \frac{q+1}{p+1} &\geqslant 1 +
\frac{p}{2q+1} + \frac{q}{2p+1} + \frac{1}{p+q+1};\cr
...
3
votes
0answers
360 views
Enestrom-Kakeya Theorem
The Enestrom-Kakeya theorem states that all roots of the polynomial:
$$p(z):=\sum_{k=0}^n a_kz^k$$
lie outside the open unit disk if the sequence $(a_k)$ is positive and decreasing.
A proof can be ...
4
votes
2answers
347 views
Could You check whether this expression is nonnegative?
I'm trying to determine if $$\bigl(x+y)^4(y+z)^4(z+x)^4 - 8x^2y^2z^2\bigl((x+y)^2 + (y+z)^2\bigr)\bigl((y+z)^2 + (z+x) ^2\bigr)\bigl((z+x)^2 + (x+y)^2\bigr) \ge0 $$
for $x,y,z>0$.
2
votes
0answers
88 views
Generalizing an approach to proving AMGM
This problem is Exercise 5.5.30 of "The Art and Craft of Problem Solving" by Paul Zeitz.
The problem asks to use the identity
$$ a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-ac-bc) $$
to prove the AMGM ...
4
votes
1answer
176 views
Polynomial inequality
I found the following problem on a website and would be curious to find a solution.
Let $a_1\ge a_2\ge\cdots\ge a_n$ be real numbers such that for all integer $k>0$:
$$a_1^k+a_2^k+\cdots+a_n^k\ge ...
1
vote
1answer
80 views
Greater than zero?
I need to show that
$$\sum_{i=k^*}^K\binom{K}{i}a^{i-1}(1-a)^{K-i-1}(i-aK)>0$$
given $K\geq k^*$, $0<a<1$ and $K$, $k^*\in\mathbb{Z^+/1}$.
I did some computer simulation and saw that it ...
5
votes
2answers
168 views
Proving a polynomial inequality over the real numbers. $x^{16}-x^{11}+x^6-x+1>0$
Prove that $x^{16}-x^{11}+x^6-x+1>0$ for $x\in R$.
So I thought of something like this:
$$x^{10}(x^6-x)+x^6-x>-1$$
$$(x^{10}+1)(x^6-x)>-1$$
But it seems to not be too much of help. While ...
2
votes
1answer
80 views
General bound on a polynomial's root with the largest norm
Is there a general bound on a polynomial's root with the largest norm?
When Rouche's theorem is used, it still seems that the polynomial's root with the largest norm still needs to be found if we ...
3
votes
1answer
74 views
How do we bound the cardinality of this group
Let $g$ be a positive integer.
How do I bound the number of elements of the group $Sp(2g,\mathbb{Z}/15)$?
Is there a polynomial bound in $g$, or can we not do better than exponential in $g$?
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 ...
29
votes
4answers
773 views
AM-GM-HM Triplets
I want to understand what values can be simultaneously attained as the arithmetic (AM), geometric (GM), and harmonic (HM) means of finite sequences of positive real numbers. Precisely, for what points ...
1
vote
2answers
114 views
Inequality in Complex Plane
In continuation to my previous post : Inequality in Complex Plane I'm still having a small problem with a similar inequality :
For $z$ such that: $|z|> 1$ I wish to prove:
$$1+|z|+\dots+|z^{n-1}| ...
1
vote
2answers
115 views
Inequality in Complex Plane
I'm studying numerical analysis and in the book I'm reading there is a theorem thats find a raduis such that all the roots of a polynomial $P$ (with coefficient in $\mathbb{C}$) are in the open disk ...
4
votes
1answer
243 views
Proving that a polynomial is positive
A Finnish mathematics competition asked to prove that for all $x$ we have $x^8-x^7+2x^6-2x^5+3x^4-3x^3+4x^2-4x+\frac{5}{2}\geq 0$ for all real $x$. I heard that it follows from Hilbert's problem that ...
1
vote
1answer
112 views
Bounding $p$-adic valuations in inequality
I'm developing an algorithm that comes across inequalities of the form
\begin{align*}
\operatorname{ord}_p(c(b)) > \alpha
\end{align*}
for some polynomial $c \in \mathbb{Q}[b]$, $c(b) = c_0 + c_1b + ...
7
votes
4answers
144 views
How to find the solution for $\frac{2x-3}{x+1} \leq 1$?
I have the following inequality:
$$\frac{2x-3}{x+1}\leq1$$
so, considering $x \neq -1$, I started multiplying $x+1$ both sides:
$$2x-3\leq x+1$$
then I subtracted $x$ both sides:
$$x-3\leq1$$
...
-2
votes
1answer
260 views
Polynomial problem
From http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2005exams/pdf_doc/maths_ext2_05.pdf:
Suppose that $a$ and b are positive real numbers, and let $f(x)=\frac{a+b+x}{3(abx)^{\frac13}}$ for $x ...
1
vote
1answer
297 views
Second degree polynomial inequality
Let $a,b,c,x,y,z\in\mathbb{R}$. Prove that $$ \left(\frac{ax+by+cz}{x-y}\right)^2+\left(\frac{ay+bz+cx}{y-z}\right)^2+\left(\frac{az+bx+cy}{z-x}\right)^2\geq(c-a)^2+(c-b)^2$$
1
vote
2answers
298 views
Inequality of three variable polynomial
I read that one can prove by AM-GM-inequality that for all $a,b,c\in\mathbb{R}_+$ we have that
$$11(a^6 + b^6 + c^6) + 40abc(ab^2 + bc^2 + ca^2) \ge 51abc(a^2b + b^2c + c^2a)$$
How this can be done? ...
