0
votes
3answers
40 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
163 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
32 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
84 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
17 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
23 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
34 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
32 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
125 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
132 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
89 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
104 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
138 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
65 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
143 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
113 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
67 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
154 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 $x,y,z>0$.
2
votes
0answers
105 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
202 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 ...