Questions tagged [symmetric-polynomials]

Questions on symmetric polynomials, polynomials in several variables that are invariant under permutation of the variables.

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2answers
47 views

Prove $(\frac{a}{b-c})^2+(\frac{b}{c-a})^2+(\frac{c}{a-b})^2 \geq 2$ [duplicate]

If $a, b, c$ are distinct real numbers, prove that $(\frac{a}{b-c})^2+(\frac{b}{c-a})^2+(\frac{c}{a-b})^2 \geq 2$ I thought of using AM-GM but that is surely not getting me anywhere ( Maybe some ...
3
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2answers
49 views

Prove$:$ $\sum\limits_{cyc} (\frac{a}{b+c}-\frac{1}{2}) \geqq (\sum\limits_{cyc} ab)\Big[\sum\limits_{cyc} \frac{1}{(a+b)^2}\Big]-\frac{9}{4}$

For $a,b,c$ are reals and $a+b+c>0, ab+bc+ca>0, (a+b)(b+c)(c+a)>0.$ Prove$:$ $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} -\frac{3}{2} \geqq (\sum\limits_{cyc} ab)\Big[\sum\limits_{cyc} \frac{...
5
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3answers
125 views

Prove $(a^2+b^2+c^2)^3 \geqq 9(a^3+b^3+c^3)$

For $a,b,c>0; abc=1.$ Prove$:$ $$(a^2+b^2+c^2)^3 \geqq 9(a^3+b^3+c^3)$$ My proof by SOS is ugly and hard if without computer$:$ $$\left( {a}^{2}+{b}^{2}+{c}^{2} \right) ^{3}-9\,abc \left( {a}^{3}+...
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0answers
22 views

Explicit expression for certain Schur polynomials

I am trying to find explicit expression for Schur polynomials of the form \begin{equation} s_\lambda(1,q,q^2,\dots, q^{d-1},q^{-c} ,q^{d+1}, \dots,q^{N-1})~, \end{equation} with $c\in \mathbb{Z}^+$ ...
2
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2answers
57 views

Inequality question.

Let $a,b,c>0$ with $a+b+c=1$. Show that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq 3 + 2\cdot\frac{(a^3 + b^3 + c^3)}{abc}$. Ohhhkk. So first off, $\begin{align} a^3 + b^3+ c^3 & =a^3 + b^...
4
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0answers
32 views

Using symmetric polynomials to find the discriminant of $x^4 + px + q$ over $\mathbb{Q}$

I'm trying to prove that the discriminant of $x^4 + px + q$ over $\mathbb{Q}$ is $-27p^4 + 256q^3$, where we define the discriminant to be $$ \Delta_f = \prod_{i < j}(\alpha_i - \alpha_j)^2 $$ I ...
1
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1answer
48 views

$\min\left(\sum_{\text{sym}}\frac{(2x^2+y)(4x + y^2)}{\underset{\ne0}{(2x + y - 2)^2}} - 3(\underset{\gt0}{x}+\underset{\gt0}{y})\right)=?$

Given positives $x$ and $y$ such that $2x + y \ne 2$ and $x + 2y \ne 2$, calculate the minimum value of $$\frac{(2x^2 + y)(4x + y^2)}{(2x + y - 2)^2} + \frac{(2y^2 + x)(4y + x^2)}{(2y + x - 2)^2} - 3(...
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0answers
9 views

Symmetric functions for multidimensional variables

I have $N$ variables (let's call them $X$) of dimensionality $D$, that I want to symmetrize. For $D=1$ I know I can use e.g. the elementary symmetric polynomials to accomplish this. What if $D>1$...
1
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3answers
37 views

Writing explicitly $(s^2-1)^2+(t^2-1)^2$ as a polynomial in $st$ and $s+t$?

Consider the symmetric polynomial $$ P(s,t)=(s^2-1)^2+(t^2-1)^2.$$ How can we write $P$ as a polynomial in the variables $st,t+s$? The Fundamental theorem of symmetric polynomials implies this is ...
1
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2answers
59 views

Proving inequality by SOS.

For $x,y,z>0.$ Prove$:$ $$P={x}^{4}y+{x}^{4}z+3\,{x}^{3}{y}^{2}-11\,{x}^{3}yz+3\,{x}^{3}{z}^{2}+3 \,{x}^{2}{y}^{3}+3\,{x}^{2}{y}^{2}z+3\,{x}^{2}y{z}^{2}+3\,{x}^{2}{z}^{ 3}+x{y}^{4}-11\,x{y}^{3}z+3\,...
1
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1answer
49 views

Find the stronger inequality of $\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ca+2b^{2}+2b}\geq \frac{1}{\sum ab}$

For $a,b,c>0$ and $a+b+c=1.$ Prove$:$ $$\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ca+2b^{2}+2b}\geqq \frac{1}{ab+bc+ca}$$ This inequality is easy and there are two nice proof by AM-GM ...
1
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1answer
69 views

Prove Newton's Identities using the properties of a symmetric polynomial.

Apologies if I didn't explain this properly in the title. I understand how we progress from (1) down, but I don't understand how to use the coefficients of EQ 5 to prove Newton's Identities. ...
7
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2answers
91 views

Prove $\frac{x^2+yz}{\sqrt{2x^2(y+z)}}+\frac{y^2+zx}{\sqrt{2y^2(z+x)}}+\frac{z^2+xy}{\sqrt{2z^2(x+y)}}\geqq 1$

For $x,y,z>0$ and $\sqrt{x} +\sqrt{y} +\sqrt{z} =1.$ Prove that$:$ $$\frac{x^2+yz}{\sqrt{2x^2(y+z)}}+\frac{y^2+zx}{\sqrt{2y^2(z+x)}}+\frac{z^2+xy}{\sqrt{2z^2(x+y)}}\geq 1$$ My solution$:$ Let $x=...
2
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3answers
72 views

Prove $(a+b+c)^3 (a+b-c)(b+c-a)(c+a-b) \leqq 27a^2 b^ 2 c^2$

For $a,b,c>0$$,$ prove$:$ $$(a+b+c)^3 (a+b-c)(b+c-a)(c+a-b) \leqq 27a^2 b^ 2 c^2$$ My proof by S-S method$,$ see here. Another proof by $pqr$ method$:$ Let $p=a+b+c,\,q=ab+bc+ca,\, r=abc.$ This ...
5
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4answers
122 views

Find minimal value of $\left(2-x\right)\left(2-y\right)\left(2-z\right)$

Let $x,y,z>0$ such that $x^2+y^2+z^2=3$. Find minimal value of $$\left(2-x\right)\left(2-y\right)\left(2-z\right)$$ I thought the equality occurs at $x = y = z = 1$ (then it is easy), but the fact ...
4
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2answers
101 views

Prove the following inequality $\sum_{i<j<k}\frac{a_ia_ja_k}{(n-2)(n-1)n}\le \bigg(\sum_{i<j}\frac{a_ia_j}{(n-1)n}\bigg)^2+\frac{1}{12}$

For positive integer $n \ge 3$, prove the following inequality $$\sum_{i<j<k}\frac{a_ia_ja_k}{(n-2)(n-1)n}\le \bigg(\sum_{i<j}\frac{a_ia_j}{(n-1)n}\bigg)^2+\frac{1}{12}$$ where $a_1+a_2+\...
1
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0answers
11 views

Combinatorics for exterior power for arbitrary Specht module

The exterior powers of the standard representation are easily seen to be the representations whose Young diagrams have only boxes in the first row or first column. But, what if I start with an ...
3
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4answers
93 views

Prove $\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b} +\frac{81abc}{4(a+b+c)^2} \geqq \frac{7}{4} (a+b+c)$

For $a,b,c>0$. Prove that$:$ $$\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b} +\frac{81abc}{4(a+b+c)^2} \geqq \frac{7}{4} (a+b+c)$$ My proof: We have$:$ $$\text{LHS}-\text{RHS} =\frac{g(a,b,c)}{4abc(a+b+...
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0answers
18 views

Proof of polynomial kernel is a kernel

Given that $k_1,\ldots,k_n: \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$ are kernels. Let $c_1, \dots, c_n \in \mathbb{R}^+$ and $p \in \mathbb{N}$. We have to prove in the following function $k$ ...
3
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3answers
61 views

Proving $(a+b+c)^2\prod_{cyc}(a+b)-4\sum_{cyc}(a^2b+a^2c)\sum_{cyc}ab\geqq 0$

From Mr. Michael Rozenberg solution: For $a,b,c>0$$,$ prove that$:$ $$(a+b+c)^2\prod_{cyc}(a+b)\geq4\sum_{cyc}(a^2b+a^2c)\sum_{cyc}ab,$$ I found two SOS proof: 1) $$\text{LHS-RHS}={\frac { \left( ...
1
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0answers
18 views

How to take the inner product of a Schur Function and Skew-Schur function quickly

Consider the Hall-inner product of a Schur Function with a Skew-Schur function. Say for instance $\langle s_{(5,3,2,1)/(3,2,1)} , s_{(3,2)} \rangle$. What is a fast way to compute this? Since $\...
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0answers
9 views

Complete homogeneous symmetric functions

I'm getting trouble to understand a definition and I hope you could help. We define $m_\lambda(x)=\sum_{a}x^a$,with $a$ varying in the set of rearrangements of the length $n$ vector with $x=(x_1,x_2,.....
4
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4answers
109 views

Polynomial $x^3-2x^2-3x-4=0$

Let $\alpha,\beta,\gamma$ be three distinct roots of the polynomial $x^3-2x^2-3x-4=0$. Then find $$\frac{\alpha^6-\beta^6}{\alpha-\beta}+\frac{\beta^6-\gamma^6}{\beta-\gamma}+\frac{\gamma^6-\alpha^6}{\...
1
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1answer
46 views

If $a,b,c>0$ and $a+b+c=1$, then prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le 3+2\cdot\frac{a^3+b^3+c^3}{abc}$.

Question: If $a,b,c>0$ and $a+b+c=1$, then prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le 3+2\cdot\frac{a^3+b^3+c^3}{abc}$. Solution: Observe that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le 3+2\...
2
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1answer
137 views

How do I show that $\sum_{\text{cyc}}\frac{3a^2-2ab+3b^2}{(a+b)^2} < \frac{9}{4}(\frac{a}{c} + \frac{c}{a}) - \frac{3}{2}$?

Let the real numbers $a,b,c \in \mathbb(0, \infty\ )$, $a\leq b\leq c$. Prove that $$\frac{3a^2-2ab+3b^2}{(a+b)^2}+\frac{3b^2-2bc+3c^2}{(b+c)^2}+ \frac{3c^2-2ac+3a^2}{(c+a)^2} \leq\frac{9}{4}(\frac{a}{...
0
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2answers
99 views

Prove $a^2+b^2+c^2 \geqslant \frac{9abc}{a+b+c}+2(1+\sqrt 2)(a-b)(b-c)$

Let $a,b$ and $c$ are positive real numbers. Prove that $$a^2+b^2+c^2 \geqslant \frac{9abc}{a+b+c}+2(1+\sqrt 2)(a-b)(b-c).$$ My proof is not nice. Indeed, we need to prove the inequality where $(a−...
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2answers
51 views

Relation between the roots and coefficient.

Let Let a, b and c be the roots of the equation $$x^3 +3x^2-1=0$$Then what is the value of expression $a^2b+b^2c+c^2a$. I got it done by evaluate the sum and difference of $a^2b+b^2c+c^2a$ and $ab^...
2
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1answer
55 views

Schur inequality

Show that for all positive real numbers $a$, $b$ and $c$ such that $abc=1$, the inequality $a+b+c+2a^4+2b^4+2c^4\ge \dfrac{3}{2}\left(a^2\left(\dfrac{1}{b}+\dfrac{1}{c}\right)+b^2\left(\dfrac{1}{a}+\...
2
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2answers
70 views

Prove $\frac{a^2}{(b+c)^2}+\frac{b^2}{(c+a)^2}+\frac{c^2}{(a+b)^2}+\frac{1}{4}\ge \frac{a^2+b^2+c^2}{ab+bc+ca}$

For $a,b,c>0$. Prove that: $$\frac{a^2}{(b+c)^2}+\frac{b^2}{(c+a)^2}+\frac{c^2}{(a+b)^2}+\frac{1}{4}\geqq \frac{a^2+b^2+c^2}{ab+bc+ca}$$ NguyenHuyen gave the following expression$:$ $$\sum \...
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1answer
27 views

Algorithm to compute complete homogeneous symmetric polynomials

Is there any algorithm to compute complete homogeneous symmetric polynomials efficiently? I was able to find algorithm to compute elementary symmetric polynomials. Example :- a1 = 2, a2 =3 So for ...
1
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1answer
80 views

Proving $\left|\begin{smallmatrix} b^2+c^2 & ab & ac \\ ba & c^2 +a^2 & bc\\ ca & cb & a^2+b^2\end{smallmatrix}\right|=4a^2b^2c^2$

I am attempting to prove the following expression, using elementary row and column operations. I have included the attempted solution. $$\det \begin{bmatrix} b^2+c^2 & ab & ac \\ ba & ...
0
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1answer
74 views

Prove $\frac{3}{2} +\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} \leqq \frac{a}{b}+\frac{b}{c} +\frac{c}{a}$

For $a,\,b,\,c>0$. Prove: $$\frac{3}{2} +\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} \leqq \frac{a}{b}+\frac{b}{c} +\frac{c}{a}$$ My work: After a lot of caculates, I found: $\text{RHS-LHS}=$ ...
2
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0answers
68 views

Macdonald's symmetric functions and Hall polynomials, Chapter 2, Lemma (1.7)

$\mathfrak{o}$ is a discrete valuation ring (local P.I.D), $\mathfrak{p}$ is the unique maximal ideal. $\pi$ is the generator of $\mathfrak{p}$, i.e. $\mathfrak{p}=<\pi>$. The residue field $k=\...
0
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3answers
46 views

If $x = b+c-a$, $y = c+a-b$, $z = a+b-c$, prove $x^3+y^3+z^3 - 3xyz = 4(a^3+b^3+c^3 -3abc)$

I got this problem in a book. While trying to solve this I got something like $$2(a+b+c)(a^2+b^2+c^2)$$ and can't move forward. Your help will be appreciated.
5
votes
3answers
75 views

Prove that $ a^2+b^2+c^2 \le a^3 +b^3 +c^3 $

If $ a,b,c $ are three positive real numbers and $ abc=1 $ then prove that $a^2+b^2+c^2 \le a^3 +b^3 +c^3 $ I got $a^2+b^2+c^2\ge 3$ which can be proved $ a^2 +b^2+c^2\ge a+b+c $. From here how can I ...
0
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0answers
9 views

How do you show a symmetric function can be factorized into a product of univariate functions?

I have a symmetric function given by $$f(x):=\frac{\det\left[{p+k\choose q+k}^{-1}{}_1F_1\left(\begin{matrix}q+k\\p+k\end{matrix};x_j\right)\right]_{j,k=1}^n}{\det[x_j^{k-1}]_{j,k=1}^n},$$ for some ...
3
votes
3answers
86 views

show this inequality $\sum_{cyc}\frac{1}{5-2xy}\le 1$

let $x,y,z\ge 0$ and such $x^2+y^2+z^2=3$ show that $$\sum_{cyc}\dfrac{1}{5-2xy}\le 1$$ try: $$\sum_{cyc}\dfrac{2xy}{5-2xy}\le 2$$ and $$\sum_{cyc}\dfrac{2xy}{5-2xy}\le\sum_{cyc}\dfrac{(x+y)^2}{\frac{...
3
votes
1answer
137 views

A certain composition into the elementary symmetric polynomials

Preliminaries Let $\mathbb{F}$ be a field such that $\operatorname{char}(\mathbb{F})\neq2$. Let $n$ be a non-zero natural number. Let $\mathbb{F}\left[x_1,x_2,\ldots,x_n \right]$ be a polynomial ...
1
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3answers
72 views

Prove $\Big[\sum\limits_{cyc} a(a^2+2bc)\Big]^3 \geqq 3(ab+bc+ca)^2 . \sum\limits_{cyc} a(a^2+2bc)^2$

For $a,b,c>0$, prove that: $$ \Big[\sum\limits_{cyc} a(a^2+2bc)\Big]^3 \geqq 3(ab+bc+ca)^2 . \sum\limits_{cyc} a(a^2+2bc)^2$$ BW works here, but it's very ugly! My try: Let $p=a+b+c,q=ab+bc+ca,r=...
1
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1answer
59 views

Stuck when transforming and solving this

Given abc=1 ( all positive real numbers). Prove that: $$\frac ab + \frac bc + \frac ca +3( \frac ba +\frac cb +\frac ac) \ge 2(a +b +c+\frac 1a+ \frac 1b +\frac1c)$$ My attempt: $$\frac ab + \frac ...
4
votes
3answers
126 views

Prove that for all positives $a, b$ and $c$, $(\sum_{cyc}\frac{c + a}{b})^2 \ge 4(\sum_{cyc}ca)(\sum_{cyc}\frac{1}{b^2})$.

Prove that for all positives $a, b$ and $c$, $$\left(\frac{b + c}{a} + \frac{c + a}{b} + \frac{a + b}{c}\right)^2 \ge 4(bc + ca + ab) \cdot \left(\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}\right)$$ ...
3
votes
3answers
124 views

Lagrange Method of Solving Cubic Equations

Let $K$ be a field (for simplicity, one may assume $K\subseteq\mathbb{C}$), and let $d\in K$. Denote $w=\sqrt[3]{d}$, and $\zeta=(-1+\sqrt{-3})/2$. If $f(x)=x^3+ax^2+bx+c$ is the minimal polynomial of ...
0
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0answers
21 views

A Problem on Symmetric Polynomials.

Consider the biquadratic equation $x^4+4px^3+6qx^2+4rx+s=0$. If $\alpha_{1}, \alpha_{2}, \alpha_{3}$ and $\alpha_{4}$ are the roots of this equation find— $\sum_{\text {symmetric}}\left(\alpha_{1}-\...
1
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1answer
56 views

How to analyze the equation $(x-y)^2=2\big( (x+y)-2\sqrt{xy} \big)$?

Suppose that $x,y$ are positive real numbers and that $$ (x-y)^2=2\big( (x+y)-2\sqrt{xy} \big). \tag{*}$$ Then Mathematica claims that one of the following $3$ options holds: $$1. \, \, \, x=y.$$ $...
0
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0answers
17 views

What role do symmetries play in solving a quadratic and higher degree polynomial?

I'm writing an essay on Radicals and one of my questions is on symmetries and I was hoping someone would have a bit of insight or could maybe help me reword the question so I undertsand what is being ...
1
vote
1answer
70 views

Prove that $3x^3-41x+48\leq 0$ for $x \in [\sqrt 3, \sqrt 6]$

Prove that $3x^3-41x+48\leq 0$ for $x \in [\sqrt 3, \sqrt 6]$. This is from an inequality in one of Titu Andreescu’s inequality books. More exactly, $2(a+b+c)\geq 3+\frac38(a+b)(b+c)(c+a)$ for ...
3
votes
1answer
42 views

Are there simple expressions for skew Schur polynomials corresponding to hook-shaped diagrams?

I am a physicist who has been studying (skew) Schur polynomials recently. I am particularly interested in skew Schur polynomials $s_{\lambda/\mu}$, where $\lambda$ is a representation corresponding to ...
3
votes
0answers
40 views

Symmetric polynomials; $a_1,a_2,\ldots,a_n\in\Bbb{C}$ s.t. $\prod_{i=0}^n\left(a_i^k+1\right)=1,\;\forall k\in\Bbb{N}\implies a_1=\dots =a_n=0$

Let $a_1,a_2,\ldots,a_n\in\Bbb{C}$ s.t. $\prod_\limits{i=0}^n\left(a_i^k+1\right)=1,\;\forall k\in\Bbb{N}$. Prove that $a_1=a_2=\dots=a_n=0$. I have tried to show that all symmetric elementary ...
2
votes
1answer
63 views

Expansion of $(x+y)^n+(x+z)^n+(y+z)^n-x^n-y^n-z^n$ in terms of elementary symmetric polynomials

Consider the symmetric polynomial in $3$ variables $$ f_n(x,y,z)=(x+y)^n + (x+z)^n+(y+z)^n - x^n-y^n-z^n $$ where $n\geq 0$ is an integer. I'm inquiring if there is a closed formula for the ...
1
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1answer
60 views

For $a,b,c\ge0$, show $a^2 + b^2 + c^2 = 3$ implies $(a^3 + b^3 + c^3)^2 \geq 3 + 2(a^4 + b^4 + c^4)$.

$a,b,c$ are nonnegative real numbers. If $a^2 + b^2 + c^2 = 3$, prove that $$(a^3 + b^3 + c^3)^2 \geq 3 + 2(a^4 + b^4 + c^4).$$ I tried using $(a^3 + b^3 + c^3)^2 >= 3(a^3 + b^3 + c^3)abc$ to make ...

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