Inequality proof by using the Muirhead inequality.

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### 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 ...
1 vote
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### Proving Muirhead’s inequality by AM-GM

I don't understand the second proof of Muirhead’s inequality in page 10-11 First prove \eqref{constantMuirhead} using AM-GM: Let $(c_i)_{i=1}^n$ be a sequence of real numbers such that $c_i \neq 0$ ...
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### a,b,c>0 and prove $\frac{a}{a+\sqrt{(a+2b)(a+2c)}}+\frac{b}{b+\sqrt{(b+2a)(b+2c)}}+\frac{c}{c+\sqrt{(c+2b)(c+2a)}}\le \frac{3}{4}.$

Let $a,b,c>0$. Prove that $$\frac{a}{a+\sqrt{(a+2b)(a+2c)}}+\frac{b}{b+\sqrt{(b+2a)(b+2c)}}+\frac{c}{c+\sqrt{(c+2b)(c+2a)}}\le \frac{3}{4}.$$ It is from a book. My tries did not lead to anything ...
1 vote
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### Conditions for Muirhead's inequality hold for cyclic sums

I know that Muirhead's inequality apply only for symmetrical sums, but all inequalities with cyclic sums I have seen have the sequence in the greater side majorizing the sequence in the smaller side (...
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### Conjecture for the inequality $f\left(x\right)+f\left(y\right)+f\left(z\right)\ge1$ where $f\left(x\right)=\frac{x}{x+\frac{1}{x}+1}$ which seems easy

I come back with an inequality checked with Desmos : Let $x\in R^*$ then define : $$f\left(x\right)=\frac{x}{x+\frac{1}{x}+1}$$ Then do we have : $$f\left(x\right)+f\left(y\right)+f\left(z\right)\ge1$$...
1 vote
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### $\sum \frac{a}{b+c+d}\le \frac{2\sum a^2}{\sum ab}$ if $\sum a =4$

Let $a, b, c, d$ positive real numbers such that $a+b+c+d=4$. Prove that $$\frac{a}{b+c+d}+\frac{b}{c+d+a}+\frac{c}{d+a+b}+\frac{d}{a+b+c}\le \frac{2(a^2+b^2+c^2+d^2)}{ab+ac+ad+bc+bd+cd}.$$ My idea is ...
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Let $\ (x,y,z) \$ be positive real numbers such that $\ \ (x+y)(y+z)(z+x)=8 \ \$ Prove That $x^3y^3+y^3z^3+z^3x^3+x^2y^2z^2-4xyz>=0$ Muirhead Sols First homogenize, by multiplying by $2$ and ...
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### Find the largest $t$ such that for all positive $x, y, z$ the following inequality is satisfied

Find the largest $t$ such that for all positive $x, y, z$ the following inequality is satisfied: $(xy+xz+yz) \left(\dfrac{1}{x+y}+\dfrac{1}{x+z}+\dfrac{1}{y+z}\right)^2 \geq t$. If there were such an ...
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### Prove the inequality $\displaystyle\sum_{s y m} x^{4} y^{2} z \geqslant 2 \sum_{s y m} x^{3} y^{2} z^{2}$

Prove for positive $x,y,z$ the inequality $x^{4} y^{2} z+x^{4} z^{2} y+y^{4} x^{2} z+y^{4} z^{2} x+z^{4} y^{2} x+z^{4} x^{2} y \geqslant 2(x^{3} y^{2} z^{2}+x^{2} y^{3} z^{2}+x^{2} y^{2} z^{3})$. I ...
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1 vote
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### For real numbers $x>0, y>0, z>0$ and $x y z=1 .$ prove that $x^{6}+y^{6}+z^{6} \geq x^{5}+y^{5}+z^{5}$

For real numbers $x>0, y>0, z>0$ and $x y z=1 .$ Prove that $$x^{6}+y^{6}+z^{6} \geq x^{5}+y^{5}+z^{5}$$ How to show this? I tried using https://en.wikipedia.org/wiki/Muirhead%...
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### Proof verification for $x^{10}+y^{10}+z^{10}\ge x^9+y^9+z^9$ (where $xyz=1$ and $x,y,z\in \mathbb{R}^+$)

My teacher has shown me the following problem: Problem. Let $x,y,z\in \mathbb{R}_+$ with $xyz=1$. Show that:$$x^{10}+y^{10}+z^{10}\ge x^9+y^9+z^9.$$ I think I solved the problem using Muirhead's ...
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