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Questions tagged [uvw]

This is a very useful method for the proof of polynomial inequalities with three variables. Sometimes it works for more variables.

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UVW method application of basic theorem

Find the minimum and maximum value of $x+y+z+xy+yz+xz$ if $x^2+y^2+z^2 = 1$ I converted it to uvw, $3u+3v^2$ is the expression, $(3u)^2-2(3v^2)=1$, is the constraint. now I don't know what to do, ...
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1answer
125 views

show that $xyz(x-y)(x-z)(y-z)\le 27$

Let $\{x,y,z\}\subset[0,+\infty)$,and $x+y+z=6$. Show that: $$xyz(x-y)(x-z)(y-z)\le 27$$ I tried AM -GM but without success. $$xyz\le\left(\dfrac{x+y+z}{3}\right)^3=8$$ maybe $$(x-y)(x-z)(y-z)\le \...
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1answer
117 views

Square root inequality cyclic

Is it true that if $a+b+c=1$, where $a$, $b$ and $c$ are nonnegative real numbers, then $$(3-2\sqrt{2})\sum\limits_{cyc}\sqrt{ab}+2\sqrt{2}-1\geq\sum\limits_{cyc}\sqrt{(1-a)(1-b)}?$$ Edit: I was told ...
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0answers
167 views

Prove $\frac{x}{y}+ \frac{y}{z}+ \frac{z}{x}\geqq \frac{2\,x}{y+ z}+ \frac{2\,y}{z+ x}+ \frac{2\,z}{x+ y}$ for $x,\,y,\,z\in [\,j,\,k\,j\,]$

For $x,\,y,\,z\in [\,j,\,k\,j\,]$ $$\frac{x}{y}+ \frac{y}{z}+ \frac{z}{x}\geqq \frac{2\,x}{y+ z}+ \frac{2\,y}{z+ x}+ \frac{2\,z}{x+ y}$$ is true with $j= constant,\,k= constant> 8$ and $k_{\,\max}$...
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0answers
58 views

Prove $\sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,a+ b\,)}}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,c+ a\,)}}$ with $a,\,b,\,c> 0$

Let $a,\,b,\,c$ be positive numbers. Prove that $$\sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,a+ b\,)}}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,c+ a\,)}}$$ I tried Holder and $\lceil$ https://...