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Questions tagged [symmetric-polynomials]

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

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Proof of Newton's Formulas.

This is question 22 in section 14.6 in Dummit and Foote, I am trying to understand its solution: (Newton's Formulas)Let $f(x)$ be a monic polynomial of degree $n$ with roots $\alpha_1, \dots, \alpha_n$...
Intuition's user avatar
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4 votes
0 answers
36 views

Relation between Fourier series and Schur polynomials

I would like to know how to express the Fourier series of a symmetric function, $f(\theta_1,...,\theta_N)$, in terms of Schur polynomials $s_\lambda(x_1,...,x_N)$ in the variables $x_j=e^{i\theta_j}$. ...
thedude's user avatar
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3 votes
0 answers
90 views

Galois group of a quartic, determine all intermediate subfields explicitly

Let $F$ be the splitting field of an irreducible quartic polynomial $f \in \Bbb Q[x]$. If Galois group of $F/\Bbb Q$ is $D_4$, I try to determine all intermediate subfields explicitly. $D_4=⟨σ,τ⟩$, $σ=...
hbghlyj's user avatar
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1 answer
43 views

Find the dimension and basis of the vector of the symmetric polynomials

Let $V$ be the vector space of polynomials in two variables $x$ and $y$ over $\mathbb{R}$ with degree at most two. That is, $V = \{a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 : a_i \in \mathbb{R}\}$. ...
Shiloh Dynastie's user avatar
2 votes
0 answers
28 views

Finding the sum of the floor function of $a,(b-1)/2,c$ given two symmetric sums

Problem: Let $a<b<c$ be $3$ real numbers satisfying $a+b+c=6$, $ab+bc+ca=9$. Then, determine the value of $\lfloor{a}\rfloor+\lfloor\frac{b-1}{2}\rfloor+\lfloor{c}\rfloor$. My method of solution:...
Cognoscenti's user avatar
1 vote
0 answers
30 views

Sums of characters over over partitions of equal length

Let $\chi^{\lambda}$ and $\chi^{\mu}$ be irreducible characters of the symmetric group $S_n$. Their inner product satisfies $\langle \chi^{\lambda}, \chi^{\mu}\rangle =\sum_{\nu} \frac{1}{z_{\nu}} \...
Andrew's user avatar
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1 answer
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Solving system of power sum symmetric polynomials

I'm interested in solving the following system of $n$ equations in the unknowns $x_i, i=1, ..., n$ $$ \sum_{i=1}^n x^k_i = \alpha_k$$ where $k=1, ..., n$. The LHS is the power sum symmetric polynomial ...
blundered_bishop's user avatar
0 votes
0 answers
14 views

Isometric automorphisms of the ring of symmetric functions

I was trying to understand how special the $\omega$ involution on the ring of symmetric functions $\Lambda$ or $\Lambda^n$ (restriction to $n$ variables, just in case if by some magic, the situation ...
yeetcode's user avatar
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1 vote
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Monomial symmetric polynomial as linear combination of power sum polynomials

I'm trying to understand the formula for $E_{\lambda\mu}$ given in proposition 2.13 of this paper (page 38). I don't understand what is meant by $\nu^{(1)} \cup \nu^{(2)} \cup \cdots \cup \nu^{(\ell(\...
Stéphane Laurent's user avatar
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17 views

Skew Jack polynomial

I found the following formula in an article for the skew Jack polynomials: $$ J_{\lambda\setminus\mu} = \sum_\nu\frac{\langle J_\lambda, J_\mu J_\nu\rangle}{\langle J_\nu, J_\nu\rangle}J_\nu. $$ But ...
Stéphane Laurent's user avatar
1 vote
2 answers
87 views

How to prove $\sum_{i=1}^n\frac{(1-a_i)^n}{a_i\prod_{j\neq i}(a_j-a_i)}=\frac{1}{a_1\cdots a_n}-1$?

Prove that $\displaystyle\sum_{i=1}^n\frac{(1-a_i)^n}{a_i\prod_{j\neq i}(a_j-a_i)}=\frac{1}{a_1\cdots a_n}-1$ for distinct $a_1,\cdots,a_n\in\mathbb{R}\backslash\{0\}$. I noticed that it is equivalent ...
Orange Soda's user avatar
1 vote
1 answer
107 views

How many possible values of $k$ are there?

I considered this problem, but, I cannot solve. $a,b,c$ are a real numbers with $(a,b,c)\neq (0,0,0)$. $k$ is defined as follows. $$k=\frac{(a^2+b^2+c^2)(a^3+b^3+c^3)}{(a^5+b^5+c^5)}$$ if $ab+bc+ca=0$,...
user1287291's user avatar
0 votes
0 answers
16 views

Laplace-Beltrami operator on the set of symmetric homogeneous polynomials

I've read in a paper that the Laplace-Beltrami operator on the set of symmetric homogeneous polynomials is defined by $$ \frac{\alpha}{2}\sum_{i=1}^m\frac{\partial^2}{\partial y_i^2} + \sum_{1\leq i \...
Stéphane Laurent's user avatar
1 vote
0 answers
54 views

Calculate the discriminant of $X^n+aX^{n-1}+b$, given the discriminant of $X^n+aX+b$

Calculate the discriminant of $X^n+aX^{n-1}+b \in \mathbb{Z}[X]$, knowing that the discriminant of $X^n+aX+b$ is $(-1)^{\frac{n^2+n-2}{2}}(n-1)^{n-1}a^n+(-1)^{\frac{n(n-1)}{2}}n^nb^{n-1}$. All I have ...
Valere's user avatar
  • 1,255
1 vote
0 answers
18 views

Algorithm to create a polynomial invariant only under specific permutations of the variables

I was solving the following problem (1.2.10 from Dixon and Mortimer's Permutation Groups): Given the group $G =\langle(x_1,x_2, x_3, x_4),(x_1,x_3) \rangle$, give an example of a polynomial that's ...
Robert Lee's user avatar
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1 vote
3 answers
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What is the value of $a^2+b^2+c^2+d^2$ and $a^3+b^3+c^3+d^3$ for $a,b,c,d$ roots of $f(x)=(x-1)^2(x-2)^2+1$

My idea was $f(x)=0 \iff (x-1)^2(x-2)^2=-1$, so $(x-1)(x-2)=i$ but this leads to solutions of the form $x=\frac{1}{2}(3\pm \sqrt{1\pm 4i})$. Now I could tediously calculate the sum of squares by hand, ...
Octot's user avatar
  • 177
1 vote
0 answers
57 views

Generating Function for Modified Multinomial Coefficients

The multinomial coefficients can be used to expand expressions of the form ${\left( {{x_1} + {x_2} + {x_3} + ...} \right)^n}$ in the basis of monomial symmetric polynomials (MSP). For example, $$\...
Bear's user avatar
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0 votes
1 answer
107 views

Galois theory and Lagrange's method on quintics

I have just begun studying Galois Theory from Stewart's book and got some questions with some conclusions regarding Lagrange's methods when discussing the quintics. For reference, I will put below ...
HJE's user avatar
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1 vote
1 answer
70 views

Prove $\sigma_{n-1}\ge2n-n^2+(n-1)\sigma_1$ for $(x_1\dots,x_n)\in[0,1]^n$

Let $x_1$, $x_2\dots$, $x_n\in[0,1]$, prove that $$\sum_{k=1}^n\prod_{i\ne k}x_i\ge2n-n^2+(n-1)\sum_{k=1}^nx_k.\tag1$$ There is an obvious proof using linearity in each variable, by which we may ...
youthdoo's user avatar
  • 1,271
1 vote
2 answers
112 views

Coefficient of polynomial of $x^k$

Consider a polynomial of power n: $P(x)=1+x+x^2+\dots+x^n$ How do I find coefficient of $x^k$, where $0\le k\le 3n$ of the polynomial $P^3(x)$? I have tried plugging in different values of $n$ to find ...
JavaGamesJAR's user avatar
1 vote
0 answers
29 views

Express polynomial in terms of elemetary symmetric functions

I'm working independently through the textbook Algebra by Cohn and have reached the section on elementary symmetric functions. I thought I had understood them but am confused by this question: Express ...
Nathan's user avatar
  • 11
0 votes
1 answer
43 views

Proving the uniqueness of the representation of the fundamental theorem on symmetric polynomials.

I am having trouble reconciling myself with a solution presented in the text Galois Theory by Harold M. Edwards. The question is presented here. For context, the symmetric functions $\sigma_1, \...
ji__ka__'s user avatar
3 votes
3 answers
156 views

Proving $5(a^2+b^2+c^2)+ab+bc+ca\ge 2\sum_{cyc}a\sqrt{5a^2+b^2+c^2+ab+ac}.$

For all $a,b,c\ge 0$ prove that$$\color{black}{5(a^2+b^2+c^2)+ab+bc+ca\ge 2\sum_{cyc}a\sqrt{5a^2+b^2+c^2+ab+ac}.}$$ I've tried to use AM-GM without success. Indeed, $$3\cdot RHS=2\sum_{cyc}3a\sqrt{5a^...
Sickness's user avatar
0 votes
2 answers
126 views

$a+b+c−3≥k(a−b)(b−c)(c−a)$

Given that $a,b,c \ge 0$ satisfy $ab+bc+ca=3$. Find the maximum value of the real number $k$ such that the following inequality is always true:$$a+b+c-3 \ge k(a-b)(b-c)(c-a)$$ "I got $c=0,ab=3 \...
POQ123's user avatar
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2 votes
2 answers
123 views

Proving $\sum_{cyc}\frac{a}{\sqrt{8a+bc}}\le \frac{\sqrt{a+b+c+abc}}{2}$ if $ab+bc+ca=3.$

Problem. Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=3.$ Prove that$$\color{black}{\frac{a}{\sqrt{8a+bc}}+\frac{b}{\sqrt{8b+ca}}+\frac{c}{\sqrt{8c+ab}}\le \frac{\sqrt{a+b+c+abc}}{2}.}$...
TATA box's user avatar
2 votes
1 answer
83 views

Proving $\sqrt{31a+b+c} +\sqrt{31b+a+c} +\sqrt{31c+b+a}\le 3\sqrt{3}\cdot\sqrt{a+b+c+8}$? when $a+b+c+abc=4.$

I came up with the inequality accidentally so there is no original proof so far. It would be great if you can give some useful help to prove it. Problem. Given non-negative real numbers $a,b,c$ ...
TATA box's user avatar
1 vote
0 answers
50 views

Proving by induction a result relating to elementary symmetric functions

Let $f(x_1,...,x_n)$ be a symmetric polynomial, so that $f(x_{\sigma(1)},...,x_{\sigma(n)}) = f(x_1,...,x_n)$ for each $\sigma \in S_n$. For $1 \leq k \leq n$, we denote by $s_k$ the $k^{\text{th}}$ ...
Menander I's user avatar
1 vote
2 answers
135 views

Find a nice proof for an elegant problem.

Question. For any $a,b,c>0$ then prove that$$\frac{\sqrt{5a^2+4bc}}{bc}+\frac{\sqrt{5b^2+4ca}}{ca}+\frac{\sqrt{5c^2+4ab}}{ab}\le \frac{9}{4}\left(\frac{a^2+b^2+c^2}{abc}+\frac{a+b+c}{ab+bc+ca}\...
Dragon boy's user avatar
0 votes
1 answer
83 views

Inequality sum of square root $3a^3+bc$ when $a^2+b^2+c^2=3$.

Remark. I really like inequality problems, but I've found that I have severe difficulties with ones that are very elegant form. I can prove something simple like this $$\color{black}{\sqrt{3a+bc}+\...
Sickness's user avatar
-2 votes
1 answer
97 views

How to solve $\sqrt{\frac{a}{bc+2}}+\sqrt{\frac{b}{ca+2}}+\sqrt{\frac{c}{ab+2}}\le \sqrt{\frac{3}{2}}\cdot\sqrt{a+b+c-1}$? [closed]

Let $a,b,c\ge 0: ab+bc+ca=3.$ Prove that $$\color{black}{\sqrt{\frac{a}{bc+2}}+\sqrt{\frac{b}{ca+2}}+\sqrt{\frac{c}{ab+2}}\le \sqrt{\frac{3}{2}}\cdot\sqrt{a+b+c-1}.}$$ Equality holds at $a=b=c=1.$ ...
Dragon boy's user avatar
0 votes
1 answer
165 views

Inequality $\frac{1}{\sqrt{ab+2}}+\frac{1}{\sqrt{bc+2}}+\frac{1}{\sqrt{ca+2}}\ge \frac{2\sqrt{3}}{\sqrt{a+b+c+abc}}.$

Let $a,b,c\ge 0: ab+bc+ca=3.$ Prove that $$\frac{1}{\sqrt{ab+2}}+\frac{1}{\sqrt{bc+2}}+\frac{1}{\sqrt{ca+2}}\ge \frac{2\sqrt{3}}{\sqrt{a+b+c+abc}}.$$ Equality occurs when $a=b=c=1.$ Also, there's an ...
Dragon boy's user avatar
2 votes
1 answer
74 views

Inequality $\frac{1}{\sqrt{4a^2+4b^2+17ab}}+\frac{1}{\sqrt{4b^2+4c^2+17bc}}+\frac{1}{\sqrt{4c^2+4a^2+17ca}}\ge \frac{3}{5}. $

Let $a,b,c\ge 0: ab+bc+ca+abc=4.$ Prove that $$\color{black}{\frac{1}{\sqrt{4a^2+4b^2+17ab}}+\frac{1}{\sqrt{4b^2+4c^2+17bc}}+\frac{1}{\sqrt{4c^2+4a^2+17ca}}\ge \frac{3}{5}. }$$ Equality holds at $a=b=...
Sickness's user avatar
0 votes
3 answers
71 views

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

Let $a,b,c>0: abc=1.$ Prove that$$\frac{4}{(a+1)(b+1)(c+1)}+\frac{1}{4}\ge \frac{a}{(a+1)^2}+\frac{b}{(b+1)^2}+\frac{c}{(c+1)^2}.$$ I've tried to use equivalent steps but it is quite complicated. ...
Anonymous's user avatar
0 votes
1 answer
72 views

Prove $\frac{bc}{\sqrt{7a+9}}+\frac{ca}{\sqrt{7b+9}}+\frac{ab}{\sqrt{7c+9}}\le \frac{3}{4}; a+b+c=3.$ [closed]

Given non-negative real numbers $a,b,c$ satisfying $a+b+c=3.$ Prove that $$\frac{bc}{\sqrt{7a+9}}+\frac{ca}{\sqrt{7b+9}}+\frac{ab}{\sqrt{7c+9}}\le \frac{3}{4}.$$ Equality holds at $a=b=c=1$ or $a=b=\...
Sickness's user avatar
0 votes
1 answer
56 views

Prove $\frac{bc}{\sqrt{3a+1}}+\frac{ca}{\sqrt{3b+1}}+\frac{ab}{\sqrt{3c+1}}\le \frac{3}{2}$ if $a^2+b^2+c^2=3.$

Given non-negative real numbers $a,b,c$ satisfying $a^2+b^2+c^2=3.$ Prove that $$\color{black}{\frac{bc}{\sqrt{3a+1}}+\frac{ca}{\sqrt{3b+1}}+\frac{ab}{\sqrt{3c+1}}\le \frac{3}{2}.}$$ Here is my ...
Dragon boy's user avatar
5 votes
3 answers
160 views

Prove $2(a+b+c)\left(1+\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\ge 3(a+b)(b+c)(c+a)$ for $abc=1.$

Let $a,b,c>0: abc=1.$ Prove that: $$2(a+b+c)\left(1+\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\ge 3(a+b)(b+c)(c+a). $$ I've tried to use a well-known lemma but the rest is quite complicated for me. ...
Anonymous's user avatar
1 vote
1 answer
145 views

How to prove $\frac{1}{\sqrt{14a^2+b^2+c^2}}+\frac{1}{\sqrt{14b^2+a^2+c^2}}+\frac{1}{\sqrt{14c^2+b^2+a^2}}\ge \frac{9}{4(a+b+c)}$?

Prove that$$\frac{1}{\sqrt{14a^2+b^2+c^2}}+\frac{1}{\sqrt{14b^2+a^2+c^2}}+\frac{1}{\sqrt{14c^2+b^2+a^2}}\ge \frac{9}{4(a+b+c)},$$holds for all $a,b,c>0.$ I tried to use C-S $\dfrac{1}{x}+\dfrac{1}{...
Sickness's user avatar
1 vote
1 answer
107 views

Prove $\sqrt[3]{3a+bc}+\sqrt[3]{3b+ca}+\sqrt[3]{3c+ab}\le \frac{3}{2}\sqrt[3]{a^2+b^2+c^2+29}$ for $ab+bc+ca=3.$

If $a,b,c\ge 0: ab+bc+ca=3$ then prove $$\sqrt[3]{3a+bc}+\sqrt[3]{3b+ca}+\sqrt[3]{3c+ab}\le \frac{3}{2}\sqrt[3]{a^2+b^2+c^2+29}.$$ I tried to use AM-GM, Cauchy-Schwarz without success. Notice that $a^...
Dragon boy's user avatar
0 votes
2 answers
84 views

If $ab+bc+ca=3,$ prove $\frac{1}{\sqrt{a+b}}+\frac{1}{\sqrt{c+b}}+\frac{1}{\sqrt{a+c}}\le \sqrt{\frac{2(a+b+c)+21}{6}}.$

If $a,b,c\ge 0: ab+bc+ca=3,$ then prove $$\frac{1}{\sqrt{a+b}}+\frac{1}{\sqrt{c+b}}+\frac{1}{\sqrt{a+c}}\le \sqrt{\frac{2(a+b+c)+21}{6}}.$$ I've tried to use Cauchy-Schwarz and it's $$\frac{1}{a+b}+\...
Anonymous's user avatar
1 vote
3 answers
143 views

Prove $\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\ge \sqrt{2(ab+bc+ca)+12}$ when $a+b+c=3.$

Given non-negative real numbers $a,b,c$ satisfying $a+b+c=3.$ Prove that$$\color{black}{\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\ge \sqrt{2(ab+bc+ca)+12}.}$$ Equality holds at $a=b=c=1$ or $a=b=0;c=3.$ I ...
Dragon boy's user avatar
5 votes
1 answer
155 views

A system of polynomials has Galois group G, a subgroup of $P_n$. Why are invariant polynomials of the roots rational, can you calculate them?

I have been studying systems of equations based upon iterating a polynomial. The complete Galois group of these systems is only a subgroup of the permutation group. There are many invariant ...
tippy2tina's user avatar
8 votes
3 answers
269 views

How to prove $2\left(\sqrt{ab-1}+\sqrt{bc-1}+\sqrt{ca-1}\right)\le \left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)\sqrt{\sqrt{ab}+\sqrt{bc}+\sqrt{ca}}.$?

Question. Let $a,b,c>0: abc=a+b+c+2.$ Prove that$$2\left(\sqrt{ab-1}+\sqrt{bc-1}+\sqrt{ca-1}\right)\le \left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)\sqrt{\sqrt{ab}+\sqrt{bc}+\sqrt{ca}}.$$I am looking ...
Anonymous's user avatar
-1 votes
2 answers
128 views

Prove $\color{black}{\sqrt{\frac{7a+2}{b+c}}+\sqrt{\frac{7b+2}{a+c}}+\sqrt{\frac{7c+2}{b+a}}\ge \frac{9\sqrt{2}}{2} },$ if $a+b+c+abc=4.$

Problem. Let $a,b,c\ge 0: ab+bc+ca>0$ and $a+b+c+abc=4.$ Prove that $$\color{black}{\sqrt{\frac{7a+2}{b+c}}+\sqrt{\frac{7b+2}{a+c}}+\sqrt{\frac{7c+2}{b+a}}\ge \frac{9\sqrt{2}}{2} .}$$ Source: ...
Sickness's user avatar
0 votes
1 answer
72 views

Find minimal value $P=(a+b)(b+c)(c+a)$ if $ab+bc+ca=\sqrt{a}+\sqrt{b}+\sqrt{c}>0.$

Question. Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=\sqrt{a}+\sqrt{b}+\sqrt{c}>0.$ Find the minimal value of $P$ $$P=(a+b)(b+c)(c+a).$$ I check that Mininum is $8$ achieved at $...
Dragon boy's user avatar
1 vote
1 answer
77 views

Pieri formula for power sum symmetric polynomial

I know Pieri's formula for elementary symmetric polynomials and for complete homogeneous symmetric polynomials, but is there an analogue for power sum symmetric polynomial? It seems that it should be ...
AndrewGap's user avatar
  • 111
1 vote
2 answers
89 views

Prove $\color{black}{\frac{1}{\sqrt{a+b+2}}+\frac{1}{\sqrt{c+b+2}}+\frac{1}{\sqrt{a+c+2}}\ge1+\frac{1}{\sqrt{2(a+b+c-1)}}.}$

Question. Let $a,b,c\ge 0: a+b+c+abc=4.$ Prove that$$\color{black}{\frac{1}{\sqrt{a+b+2}}+\frac{1}{\sqrt{c+b+2}}+\frac{1}{\sqrt{a+c+2}}\ge1+\frac{1}{\sqrt{2(a+b+c-1)}}.}$$ Because equality holds at $a=...
Dragon boy's user avatar
3 votes
6 answers
474 views

How to prove $ a+b+c+\sqrt{bc}+\sqrt{ca}+\sqrt{ab}\ge 6$?

Question. Prove $$ a+b+c+\sqrt{bc}+\sqrt{ca}+\sqrt{ab}\ge 6,$$ when $a,b,c\ge 0: ab+bc+ca+abc=4.$ My idea: I've tried to use AM-GM as $$\bullet \sum \sqrt{ab}\ge 2\sum \frac{ab}{a+b}=2(ab+bc+ca)\sum \...
Sickness's user avatar
0 votes
0 answers
92 views

What are the generators of polynomials symmetric under a subgroup of the permutation group

I have been studying iterated polynomials, specifically let $P(x)=x^2+c$ and consider the equation $P(P(P(x)))=x$. After dividing out the solutions of $P(x)=x$, we have six solutions, which obey the ...
tippy2tina's user avatar
2 votes
2 answers
99 views

Prove that tan($Z$) = $\frac{e_1t - e_3t^3 + e_5t^5 - \cdots}{1 - e_2t^2 + e_4t^4 - \cdots}$

Let $e_r$ and $p_r$ denote the $r$-th elementary symmetric function and power sum, respectively. Let $t$ be a formal variable and define $$Z := p_1t-p_3t^3/3+ p_5t^5/5- \cdots $$ Prove that the ...
Eduardo4313's user avatar
2 votes
1 answer
74 views

$\sum \frac{a^2+b^2}{c^2}-\frac{7}{2} \geq \sqrt[3]{\prod \left(\frac{a}{b+c}+\frac{1}{3}\right)}$ for a,b,c real

Let: $a,b,c>0$. Prove that: $$\frac{a^2+b^2}{c^2}+\frac{b^2+c^2}{a^2}+\frac{c^2+a^2}{b^2}-\frac{7}{2}\geq \sqrt[3]{\left(\frac{a}{b+c}+\frac{1}{3}\right)\left(\frac{b}{c+a}+\frac{1}{3}\right)\left(\...
Lục Trường Phát's user avatar

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