0
votes
0answers
25 views

How to solve the inequality: $\prod_{k=1}^N\left(x^k-k^2\right)\gt0$

Given the inequality: $$\displaystyle\prod_{k=1}^N\left(x^k-k^2\right)\gt0$$ how can I solve it? I suppose there is a difference if $N=2n$ or $N=2n+1$ with $n\in\mathbb{N}$, but I'm unable to find a ...
3
votes
2answers
60 views

prove $\sum\limits_{cyc} \frac {a^3} {b+c+d} \geq \frac {1} {3}$

Show that if $a,b,c,d \geq 0$ and $ab+bc+cd+da=1$ :$$\sum\limits_{cyc} \frac {a^3} {b+c+d} \geq \frac {1} {3}$$ yet again it should be solved with Cauchy inequality. thing i have done so far: ...
0
votes
4answers
66 views

If $a^2=b^2+c^2$ and $0<n<2$ prove $a^n<b^n+c^n$

If $a^2=b^2+c^2$ and $a,b,c$ are positive real numbers, prove (a) if $n>2$ then $a^n>b^n+c^n$, (b) if $0<n<2$ then $a^n<b^n+c^n$. Part (a) was easy to prove: $a^2=b^2+c^2$ and ...
2
votes
3answers
164 views

How can this equality be established by elementary algebraic means?

Let $x \geq 1$. Then is it true that $2x^3 - 3x^2 + 2 \geq 1$? If so, how can I show this using only elementary ideas such as factorisation? Of course, I can demonstrate this using the methods of ...
0
votes
1answer
37 views

prove $\frac {x}{ay+bz} + \frac {y}{az+bx} + \frac {z}{ax+by} \geq \frac {3}{a+b}$

show that$$\frac {x}{ay+bz} + \frac {y}{az+bx} + \frac {z}{ax+by} \geq \frac {3}{a+b}$$ using Cauchy inequality i can write $$(\frac {x}{ay+bz} + \frac {y}{az+bx} + \frac {z}{ax+by}) \times ...
3
votes
0answers
85 views

Polynomial P(x) such that [closed]

Let $P(x)$ be a real polynomial with degree $n$ such that $|P(x)| \lt 1$ for all $|x| \le 1$. Prove that $P(2) \lt 4^n$.
1
vote
0answers
34 views

Decomposition of polynomials and inequality

This was asked in comment here by @23rd : If $f$ is a polynomial with $\deg f=n\ge2$, then there exist polynomials $g$ and $h$, such that $$f(x)=2xg(x)−h(x)$$ $$\deg g\le n−1, \quad \deg h \le ...
0
votes
6answers
68 views

solving the inequalty

are there any ways to solve :$ x^4 -6x^3 +28x^2 -64x +96 >0$ ?
-1
votes
6answers
49 views

Polynomial factors involving inequalities

How to factorise the polynomial $p(x) = x^4-2x^3 + 2x - 1$. Hence, solve the inequality $p(x) \gt 0$ ?
0
votes
0answers
19 views

Finding matrix index from triangular array offset

I have a mapping from a lower triangular matrix, A, to a vector,v: A(i,j) -> v( $\lfloor i(i+1)/2 \rfloor + j$ ) $i,j\in[0,N]$, $j\leq i$, $N\in\cal{N}$, $N\geq 0$ (so, my first row is row 0, and ...
0
votes
1answer
78 views

$­\prod_{k=1}^{n}(x_k-a)(x_k-b)\leqslant\sum_{q=1}^{n}x_q^2\prod_{p=1,p\neq q}^{n}(x_p-a)(x_p-b).$?

Is there a name for this formula? For $f_k,w_k\geqslant0$. $$­\prod_{k=1}^{n}f_k\leqslant\sum_{q=1}^{n}w_q\prod_{p=1,p\neq q}^{n}f_p.$$ I believe that there is $w_k$ that make the formula true. Am I ...
-3
votes
2answers
45 views

Cauchy-Schwarz Inequality [closed]

Hi,I want to prove above unequal.All elements are positive or bigger than zero. The way to prove it is not important, but because I weak in mathematical i want to know all thing about details.
6
votes
2answers
171 views

How prove this $p(x)>0$ if $p(x)=\sum_{i=0}^{n}\binom{n}{i}a_{i}x^i(1-x)^{n-i}$

let the polynomials $$p(x)=\sum_{i=0}^{n}\binom{n}{i}a_{i}x^i(1-x)^{n-i}$$ and such $$a_{0}+\sum_{a_{i}<0}(1-\dfrac{i}{n})\binom{n}{i}a_{i}>0$$ and ...
1
vote
3answers
40 views

Inequality for quartic polynomial depending on a parameter

Let $f(x) = \frac 14 x^4 - \frac \alpha2 x^2 - (\alpha-1) x - \frac \alpha 2 + \frac 3 4 $. I want to show that there exists an $\alpha>1$ such that $f(x)\geq 0 $ for $x\leq 0$. Even more, it ...
5
votes
4answers
96 views

What is the minimum value of $abc$

If the roots of the equation $$ax^2-bx+c=0$$ lie in the interval $(0,1)$, find the minimum possible value of $abc$. Edit: I forgot to mention in the question that $a$, $b$, and $c$ are natural ...
3
votes
2answers
59 views

Find the value of $\left | b-c \right |$

Given that $a, b, c \in \mathbb{Z}$, $a>10$ and $$(x-a)(x-12)+2=(x+b)(x+c)$$ Find the value of $\left | b-c \right |$ NOTE: The answer to this problem (as given on the last page of my book) is ...
3
votes
1answer
52 views

Determining equality of sets defined by polynomial inequalities in several variables

Let's say I have two sets $S_1,S_2 \subseteq \mathbb{R}^n$ each defined by a number of polynomial inequalities. Is there a computationally feasible way to find whether $S_1 = S_2$? In particular, is ...
1
vote
1answer
48 views

Finding the minimum value of a 6th degree polynomial algebraically

Is it possible to answer this question using methods of basic algebra? Find the least value of the expression $a^6 + a^4 - a^3 - a + 1$ for real value of $a$. This question is from the 2013 Philippine ...
0
votes
1answer
23 views

How to simplifying and solving this polynomial?

I have a problem with simplifying the polynomial. In the first time, I see that this polynomial is quite simple, but when I'm trying, I realized that this polynomial isn't as easy as I saw. Here is ...
0
votes
1answer
13 views

Inequalities with cubic polynomials

So I was checking my friend's notes and found this: Find the interval for which $x^3-x^2-x+1>0$ is true. We haven't been taught to factor cubic polynomials (the rest of the exercises are with ...
0
votes
1answer
40 views

Inferring a characteristic of a ratio of functions from the ratio of their derivatives

This is a strange one, but I need help trying to understand whether there is any logic behind this or not. Given $\frac {f(\sqrt{2})}{g(\sqrt{2})}=2$, and $\frac {f'(x)}{g'(x)}>2$ for all ...
0
votes
0answers
25 views

Inequalities and empty set.

Let $h_{k}(x)>0$ and $\sum_{k=1}^{l}h_{k}(x)=1$ (Here, $h_{k}(x)$ are some non-linear functions). and $\sum_{k=1}^{l}h_{k}(x)f_{k}(x)<0$ when $\sum_{k=1}^{l}h_{k}(x)g_{k}(x)=0$,$\forall ...
0
votes
1answer
17 views

Inequality Conditions

Let $h_{k}(x)>0$ and $\sum_{k=1}^{l}h_{k}(x)=1$ (Here, $h_{k}(x)$ are some continuous functions). Is the statement below correct or not? $f_{k}(x)<0$ when $g_{k}(x)=0$, $\forall x \neq 0$, ...
0
votes
1answer
38 views

Find a counter-example for inequality

Let $C>1$ be a constant. I have to find polynomial $p(t)=a_0+a_1 t+\dots +a_n t^n$ such that: $$|a_0|+|a_1|+\dots + |a_n| \le C \sup_{t\in[0,1]} |p(t)|$$ doesn't hold. Any tip?
0
votes
0answers
34 views

Finding a set of solution under some conditions

I'm looking for a set of the solution $(a,b,e)\in\mathbb R$ which satisfies the following conditions. Condition 1 : $a\lt 0$ Condition 2 : $0\lt b\lt e$ Condition 3 : ...
1
vote
2answers
98 views

How prove this here exsit $b\in R$,such $S=\{(b,b,\cdots,b)\}$,if $f(x_{1},x_{2},\cdots,x_{n})$ is the set of minimum and maximum points.

Assmue $f(x_{1},x_{2},\cdots,x_{n})$ is a second degree real polynomial with $n(n\ge 2)$ variables. Let $S$ be such that $f(x_{1},x_{2},\cdots,x_{n})$ is the set of minimum and maximum points. In ...
0
votes
1answer
35 views

Inequality between the coeficients of a quartic equation

Given the inequlity $$ ax^4-bx-c\geq 0, \quad \forall x\in \mathbb R $$ where $a, b$ and $c$ are real positive constants. Is it possible to conclude some inequality between the coefficients like ...
1
vote
1answer
33 views

Proving that $|\Phi_n(x)| > x-1$

Let $\Phi_n$ be the n-th cyclotomic polynomial. I'd like to prove that $$\forall n \geq 2, \forall x \in [2, \infty[, |\Phi_n(x)| > x-1$$ The result is clear when $n$ is prime, but I'm struggling ...
0
votes
0answers
34 views

Lower-Upper bounds on the cardinality of a bounded set

Let $S$ be a finite set which is a subset of $\{(\alpha ,\beta ):\alpha , \beta \in \mathbb{Z}, \alpha\geq 0, \beta \geq 0\}$ and $ T(x,y)=\sum_{(\alpha ,\beta ) \in S} h_{\alpha, \beta} ...
6
votes
2answers
126 views

Polynomial inequality proof

Prove $(1-a)(1-b)(1-c)(1-d)>1-a-b-c-d$ and $ a,b,c,d$ are real and between 0 and 1. I can do this with $$(1-a)(1-b)>1-a-b \\ 1-a-b+ab>1-a-b \\ ab>0 $$ But with $c$ and $d$, this ...
3
votes
1answer
133 views

A polynomial has only real roots and all coefficients $\pm 1$. Prove the degree $<4$.

Let $P(x)$ be a polynomial with only real roots and all coefficients equal to $\pm 1$. Prove that the degree of the polynomial is less than 4. This is practice for Putnam, but I am not certain where ...
1
vote
2answers
116 views

verifying a polynomial is positive on the half-line

Math people: I am running experiments that produce polynomials $P(z)$ that, in every experiment I have run, are always positive on the half-line $\{z \geq 1\}$. I want to prove analytically that the ...
0
votes
3answers
90 views

looking for help proving conjecture about the sign of a polynomial, or finding counterexample

Math people: I would like some help proving this conjecture, which is backed up by a lot of experiments, or finding a counterexample. Let $n \geq 1$ be an integer, $a_0, \ldots, a_n$ be real ...
13
votes
1answer
271 views

Annoying Polynomial Inequality

Suppose we have a polynomial satisfying $p+p''' \geq p'+p''$ for all $x$. Then $p(x)\geq 0$ for all $x$. I've been stuck on this problem for weeks. The best I can do is supposing there exists $x$ so ...
-2
votes
2answers
106 views

$x_1+x_2+\cdots+x_n\leq M$: Cardinality of Solution Set is $C(M+n, n)$

Show that the number of solutions in nonnegative integers of the inequality $$x_1+x_2+\cdots+x_n\leq M,$$ where $M$ is a nonnegative integer, is $C(M+n, n)$.
4
votes
1answer
88 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
139 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
66 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 $|z|>R$. ...
11
votes
3answers
264 views

Polynomial always non-negative

Is there an elegant way to demonstrate that (for example) $x^{2016}-1008x^2+1007\ge 0$ $\forall x\in \mathbb{R}$ ? I tried to write it as sum of squares, but I didn't succeed.
5
votes
2answers
146 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 ...)
2
votes
0answers
114 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
79 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 ...
17
votes
2answers
930 views

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
68 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
155 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, \ ...
3
votes
2answers
76 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
173 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
148 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 ...
7
votes
0answers
464 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 ...
4
votes
2answers
368 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 $x,y,z>0$.