\frac{a^2}{ b^2+c^2 } \left(\frac{b^2}{c^2+a^2}+\frac{c^2}{a^2+b^2} \right)\geq\

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\frac{a^2}{ b^2+c^2 } \left(\frac{b^2}{c^2+a^2}+\frac{c^2}{a^2+b^2} \right)\geq\

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sqing

#1 h Jul 25, 2024, 4:01 AM

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Let $a,b,c>0$ and $a\geq b+c.$ Prove that
$\frac{a^2}{ b^2+c^2 } \left(\frac{b^2}{c^2+a^2}+\frac{c^2}{a^2+b^2} \right)\geq\frac{4}{5}$ $\frac{a^2}{ b^2+c^2 } \left(\frac{b^2}{c^2+2a^2}+\frac{c^2}{2a^2+b^2} \right)\geq\frac{4}{9}$ $\frac{a^2}{ b^2+c^2 } \left(\frac{b^2}{2c^2+a^2}+\frac{c^2}{a^2+2b^2} \right)\geq\frac{2}{3}$

This post has been edited 1 time. Last edited by sqing, Jul 25, 2024, 4:09 AM

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frc

#2 h Jul 26, 2024, 3:04 AM

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We can promote it: $\frac{a^2}{b^2+c^2}(\frac{b^2}{ma^2+nc^2}+\frac{c^2}{ma^2+nb^2})\geq \frac4{4m+n}$ $\begin{aligned} L.H.S. &\geq \frac{(b+c)^2}{b^2+c^2}(\frac{b^2}{m(b+c)^2+nc^2}+\frac{c^2}{m(b+c)^2+nb^2}) \\ &\geq \frac{(b+c)^2}{b^2+c^2} \frac{(b^2+c^2)^2}{b^2[nc^2+m(b+c)^2]+c^2[m(b+c)^2+nb^2]} \\ &=\frac{(b+c)^2(b^2+c^2)}{m(b^2+c^2)(b+c)^2+2nb^2c^2} \\ &\geq\frac{(b+c)^2(b^2+c^2)}{m(b^2+c^2)(b+c)^2+\frac{n}4 (b^2+c^2)(b+c)^2} \\ &=\frac4{4m+n} \end{aligned}$

This post has been edited 1 time. Last edited by frc, Jul 27, 2024, 3:31 AM

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sqing

#3 h Jul 26, 2024, 3:37 AM

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Thanks.

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