parmenides51 wrote: Show that $(x + y + z) \big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\big) \ge 4 \big(\frac{x}{xy+1}+\frac{y}{yz+1}+\frac{z}{zx+1}\big)^2$ , for all real positive numbers $x, y $ and $z$. \begin{align*} P(t) &= (x+y+z)t^2-4(\frac{1}{y+\frac{1}{x}}+\frac{1}{z+\frac{1}{y}}+\frac{1}{x+\frac{1}{z}})t+(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\\ &=\sum (xt^2-\frac{4}{y+\frac{1}{x}}t+\frac{1}{y}) \\ &=\sum \frac{(xy+1)(t^2xy+1)-4txy}{(y+\frac{1}{x})}\\ &\geq \sum \frac{ 2\sqrt{xy} 2\sqrt{t^2xy}-4txy}{(y+\frac{1}{x})}\\ &= \sum \frac{ 4xy(|t|-t)}{(y+\frac{1}{x})}\\ &\geq 0\\ \end{align*}Then we conclude that: \[\Delta = 4(\frac{1}{y+\frac{1}{x}}+\frac{1}{z+\frac{1}{y}}+\frac{1}{x+\frac{1}{z}})^2 - (x+y+z)(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}) \leq 0\]Done!

parmenides51 wrote: Show that $(x + y + z) \big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\big) \ge 4 \big(\frac{x}{xy+1}+\frac{y}{yz+1}+\frac{z}{zx+1}\big)^2$ , for all real positive numbers $x, y $ and $z$. \begin{align*}\\ \ (\sum_{cyc} x)(\sum_{cyc} \frac{1}{x})\geq (\sum_{cyc} \sqrt{\frac{x}{y}})^2\\ \sum_{cyc} (\frac{y}{x}+1)\geq\sum_{cyc} 2\sqrt{\frac{y}{x}}\\ \sum_{cyc} \sqrt{\frac{x}{y}}\geq \sum_{cyc}\frac{2x}{xy+1}\\ \end{align*}

parmenides51 wrote: Show that $(x + y + z) \big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\big) \ge 4 \big(\frac{x}{xy+1}+\frac{y}{yz+1}+\frac{z}{zx+1}\big)^2$ , for all real positive numbers $x, y $ and $z$. Similar to above but probably different finish... \begin{align*} (x + y + z) \big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\big) &\geq (\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{z}}+\sqrt{\frac{z}{x}})^2\\ &=(\frac{x}{\sqrt{yx}}+\frac{y}{\sqrt{yz}}+\frac{z}{\sqrt{xz}})^2\\ &\geq (\frac{2x}{yx+1}+\frac{2y}{yz+1}+\frac{2z}{xz+1})^2\\ &=4(\frac{x}{yx+1}+\frac{y}{yz+1}+\frac{z}{xz+1})^2\\ \end{align*}Done.

parmenides51 wrote: Show that $$(x + y + z) \big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\big) \ge 4 \big(\frac{x}{xy+1}+\frac{y}{yz+1}+\frac{z}{zx+1}\big)^2$$, for all real positive numbers $x, y $ and $z$. Let $a,b,c$ be positive real numbers. Prove that: $$ (a+b+c) ( \frac{1}{a}+\frac{1}{b}+\frac{1}{c})\geq 9+3\sqrt[3]{\frac{(a-b)^2(b-c)^2(c-a)^2}{a^2b^2c^2}}.$$Saudi Arabia JBMO TST 2015