Inequality-2??
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Started by: hell_ever_hell_ever_
On: 1208368027|%e %b %Y, %H:%M %Z|agohover
Number of posts: 2
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Inequality-2??
hell_ever_hell_ever_ 1208368027|%e %b %Y, %H:%M %Z|agohover

Prove for positive reals a,b that

\frac {a+b}{2}-\sqrt {ab}\geq \frac {(a-b)^2(a+3b)(b+3a)}{8(a+b)(a^2+6ab+b^2)}

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unfold Inequality-2?? by hell_ever_hell_ever_, 1208368027|%e %b %Y, %H:%M %Z|agohover
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Re: Inequality-2??
computerboosycomputerboosy 1208418030|%e %b %Y, %H:%M %Z|agohover

\mbox{To prove that} \ \frac{a+b}{2}-\sqrt{ab} \geq \frac{(a-b)^2(a+3b)(3a+b)}{8(a+b)(a^2+6ab+b^2)} \\ \\ \mbox{To prove that} \ (\sqrt{a}-\sqrt{b})^2 \geq \frac{(a-b)^2(a+3b)(3a+b)}{4(a+b)(a^2+6ab+b^2)} \\ \\ \mbox{writing} \ (a-b)^2 \ \mbox{as} \ (\sqrt{a}-\sqrt{b})^2*(\sqrt{a}+\sqrt{b})^2 \\ \\ \mbox{To prove that} \ 4(a^3+b^3+7ab(a+b))\geq (3(a^2+b^2)+10ab)(a+b+2\sqrt{ab}) \\ \\ \mbox{To prove that} \ a^3+b^3+15ab(a+b)\geq2\sqrt{ab}(3a^2+3b^2+10ab) \\ \\ \mbox{On squaring the above expression ..though slightly long but still worth it} \\ \\ \mbox{To prove that} \ a^6-6a^5b+15a^4b^2-20a^3b^3+15a^2b^4-6ab^5+b^6\geq 0 \\ \\ \mbox{To prove that} \ (a-b)^6\geq0 \\ \\ \mbox{which is always true} \\ \\ \mbox{Hence proved}

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unfold Re: Inequality-2?? by computerboosycomputerboosy, 1208418030|%e %b %Y, %H:%M %Z|agohover
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