An Inequality in Triangle, with Sines

Problem

An Inequality in Triangle, with Sines

Solution

Let $\sin A=x,\,\sin B=y,\,\sin C=z.\,$ Then $0\lt x,y,z\lt 1.\,$ Define function $f:\,(0,1)\to\mathbb{R}\,$ by $\displaystyle f(t)=\ln\left(\frac{2}{t}-1\right).\,$ Clearly, $\displaystyle f''(t)=\frac{1}{t^2}-\frac{1}{(2-t)^2}\gt 0,\,$ for $t\in (0,1).\,$ Hence, by Jensen's inequality, $\displaystyle f(x)+f(y)+f(z)\ge 3f\left(\frac{x+y+z}{3}\right),\,$ implying

$\displaystyle \left(\frac{2}{x}-1\right)\left(\frac{2}{y}-1\right)\left(\frac{2}{z}-1\right)\ge \left(\frac{6}{x+y+z}-1\right)^3.$

Note that, due to the Law of Sines, equality holds only for equilateral triangles.

Acknowledgment

The problem which is due to Lorian Saceanu has been posted by Seth Easterwood at the CutTheKnotMath facebook page on 28 February. An hour later Leo Giugiuc has posted his solution.

 

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