Inequality (sin, r and s)

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sowmitra
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Inequality (sin, r and s)

Unread post by sowmitra » Wed Apr 29, 2015 9:39 pm

Prove the inequality,
\[\frac{1}{\sqrt{2\sin A}}+\frac{1}{\sqrt{2\sin B}}+\frac{1}{\sqrt{2\sin C}}\leq \sqrt{\frac{s}{r}}\]
where, $s$ and $r$ are the semi-perimeter and inradius of $\triangle ABC$.

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Nirjhor
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Re: Inequality (sin, r and s)

Unread post by Nirjhor » Thu Apr 30, 2015 11:26 pm

The inequality simplifies to \[\dfrac{1}{\sqrt{a}}+\dfrac{1}{\sqrt{b}}+\dfrac{1}{\sqrt{c}}\le \dfrac{a+b+c}{\sqrt{abc}}\Rightarrow \sqrt{ab}+\sqrt{bc}+\sqrt{ca}\le a+b+c\] after making the substitutions $2\sin A=\dfrac a R$ etc, $r=\dfrac{abc}{4sR}$, and $2s=a+b+c$.

This is a standard AM-GM result with equality at $a=b=c$ i.e. equilateral triangle.

A lower bound is $3^{3/4}$ found immediately via Jensen's, equality at equilateral triangle.
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Tahmid
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Re: Inequality (sin, r and s)

Unread post by Tahmid » Fri May 01, 2015 12:34 am

my solution is too much long :|

$\frac{1}{\sqrt{2sinA}}+\frac{1}{\sqrt{2sinB}}+\frac{1}{\sqrt{2sinC}}\leq \sqrt{\frac{s}{r}}$
or,$\sqrt{\frac{R}{a}}+\sqrt{\frac{R}{b}}+\sqrt{\frac{R}{c}}\leq \sqrt{\frac{s}{r}}$
or,$\frac{R}{a}+\frac{R}{b}+\frac{R}{c}+\frac{2R}{\sqrt{ab}}+\frac{2R}{\sqrt{bc}}+\frac{2R}{\sqrt{ca}}\leq \frac{s}{r}$
or,$R(\frac{ab+bc+ca}{abc})+2R(\frac{a\sqrt{bc}+b\sqrt{ca}+c\sqrt{ab}}{abc})\leq \frac{s}{r}$
or,$\frac{R}{abc}(ab+bc+ca)+\frac{2R}{abc}(a\sqrt{bc}+b\sqrt{ca}+c\sqrt{ab})\leq \frac{s^{2}}{\Delta }$
or,$\frac{1}{4\Delta}(ab+bc+ca)+\frac{1}{2\Delta}(a\sqrt{bc}+b\sqrt{ca}+c\sqrt{ab})\leq \frac{(a+b+c)^{2}}{4\Delta }$
or,$(ab+bc+ca)+2(a\sqrt{bc}+b\sqrt{ca}+c\sqrt{ab})\leq (a+b+c)^{2}$
or,$ab+bc+ca+2a\sqrt{bc}+2b\sqrt{ca}+2c\sqrt{ab}\leq a^{2}+b^{2}+c^{2}+2ab+2bc+2ca$
or,$2a\sqrt{bc}+2b\sqrt{ca}+2c\sqrt{ab}\leq a^{2}+b^{2}+c^{2}+ab+bc+ca$

now from AM-GM we know , $\frac{a^{2}+bc}{2}\geq \sqrt{a^{2}bc} \Leftrightarrow a^{2}+bc\geq 2a \sqrt{bc}$
samely , $ b^{2}+ca\geq 2b \sqrt{ca}$ and $c^{2}+ab\geq 2c \sqrt{ab}$

by adding then we have $2a\sqrt{bc}+2b\sqrt{ca}+2c\sqrt{ab}\leq a^{2}+b^{2}+c^{2}+ab+bc+ca$

so, we are done :)

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