Prove $\alpha+\beta+\gamma \ge \alpha\frac{\sin \beta}{\sin \alpha}+\beta\frac{\sin \gamma}{\sin \beta}+\gamma\frac{\sin \alpha}{\sin \gamma}$ Prove the inequality
$$\alpha+\beta+\gamma \ge \alpha\cdot\frac{\sin \beta}{\sin \alpha}+\beta\cdot\frac{\sin \gamma}{\sin \beta}+\gamma\cdot\frac{\sin \alpha}{\sin \gamma}$$
for any three numbers $\alpha, \beta, \gamma \in \left(0;\frac{\pi}2\right)$
My attempt:
I plan to use Rearrangement inequality. But I do not know how to start.
 A: Using Aristarchus' inequality (see here for a proof-of-Aristarchus'-inequality) which states that if $\alpha$ and $\beta$ are acute angles (i.e. between $0$ and $\pi/2$) and $\alpha<\beta$ then $\frac{\sin \beta}{ \sin \alpha}<\frac{\beta}{\alpha}<\frac{\tan \beta}{ \tan \alpha}$.
With $\alpha, \beta, \gamma \in \left(0,\frac{\pi}2\right)$ and $\alpha\le\beta\le\gamma$ gives:
\begin{align*}
(\alpha\le\beta)\quad&\quad\frac{\sin \beta}{ \sin \alpha}\le\frac{\beta}{\alpha}\quad \Rightarrow\quad\frac{\alpha}{\sin \alpha}\le\frac{\beta}{ \sin \beta}\\
(\alpha\le\gamma)\quad&\quad\frac{\sin \gamma}{ \sin \alpha}\le\frac{\gamma}{\alpha}\quad \Rightarrow\quad\frac{\alpha}{\sin \alpha}\le\frac{\gamma}{ \sin \gamma}\\
(\beta\le\gamma)\quad&\quad\frac{\sin \gamma}{\sin \beta}\le\frac{\gamma}{\beta}\quad \Rightarrow\quad\frac{\beta}{\sin \beta}\le\frac{\gamma}{ \sin \gamma}
\end{align*}
Hence putting these inequalities together:
$$\frac{\alpha}{\sin \alpha}\le\frac{\beta}{ \sin \beta}\le\frac{\gamma}{ \sin \gamma}\tag{A}$$
We also have:
$$0<\sin \alpha\le\sin \beta\le\sin \gamma<1\tag B$$
The Rearrangement Theorem states:
for every choice of real numbers
$$ x_{1}\leq \dotsb \leq x_{n}\quad \text{and}\quad y_{1}\leq \dotsb \leq y_{n}$$
and every permutation
$$x_{\sigma (1)},\dotsc ,x_{\sigma (n)}$$
of $x_1,\dotsc, x_n$ we have:
$$x_{n}y_{1}+\dotsb +x_{1}y_{n}\leq x_{\sigma (1)}y_{1}+\dotsb +x_{\sigma (n)}y_{n}\leq x_{1}y_{1}+\dotsb +x_{n}y_{n}$$
We now invoke the Rearrangement Theorem on  inequalities (A) and (B):
\begin{align*}
y_1&=\frac{\alpha}{\sin \alpha}, \quad\,\, y_2=\frac{\beta}{ \sin \beta}, \quad\,\, y_3=\frac{\gamma}{ \sin \gamma}\\
x_1&=\sin \alpha, \quad\ \,\,\, x_2=\sin \beta, \quad\ \,\,\, x_3=\sin \gamma\\
x_{\sigma (1)}&=\sin \beta, \quad x_{\sigma (2)}=\sin \gamma, \quad x_{\sigma (3)}=\sin \alpha
\end{align*}
Then, written in full this gives:
$$x_{3}y_{1}+x_{2}y_{2}+x_{1}y_{3}\leq x_{\sigma (1)}y_{1}+x_{\sigma (2)}y_{2} +x_{\sigma (3)}y_{3}\leq x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}$$
\begin{align*}
&\quad\,\,\alpha\frac{\sin \gamma}{\sin \alpha}+\beta\frac{\sin \beta}{\sin \beta}
+\gamma\frac{\sin \alpha}{\sin \gamma}\\
&\le\alpha\frac{\sin \beta}{\sin \alpha}+\beta\frac{\sin \gamma}{\sin \beta}
+\gamma\frac{\sin \alpha}{\sin \gamma}\\
&\le\alpha\frac{\sin \alpha}{\sin \alpha}+\beta\frac{\sin \beta}{\sin \beta}
+\gamma\frac{\sin \gamma}{\sin \gamma}=\alpha+\beta+\gamma
\end{align*}
Remark: for the middle inequality we could choose a different permutation, say
$$x_{\sigma(1)}=\sin \gamma, \quad x_{\sigma(2)}=\sin \alpha, \quad x_{\sigma(3)}=\sin \beta$$
to give the inequality
 $$\alpha+\beta+\gamma\ge \alpha\frac{\sin \gamma}{\sin \alpha}+\beta\frac{\sin \alpha}{\sin \beta}+\gamma\frac{\sin \beta}{\sin \gamma}$$
A: Let $f(t) = \sin t - t \cos t, 0 \leq t \leq \pi/2.$ 
We have $f(0) = 0$ and $f'(t) = \cos t - \cos t + t \sin t = t \sin t \geq 0$ if $0 \leq t \leq \pi/2.$
This implies $f(t) \geq 0$ for $0 \leq t \leq \pi/2.$
If $g(t) = \dfrac{t}{\sin t}$  then $g'(t) = \dfrac{f(t)}{\sin^2 t} \geq 0.$ So $\dfrac{t}{\sin t}$ increases on $(0,\dfrac{\pi}{2}).$
Let $ 0 < \alpha \leq \beta \leq \gamma < \pi/2$ then $ 0 < \dfrac{\alpha}{\sin \alpha} \leq \dfrac{\beta}{\sin \beta} \leq \dfrac{\gamma}{\sin \gamma}$ and $0 < \sin \alpha \leq \sin \beta \leq \sin \gamma.$
By the rearrangement inequality $ \dfrac{\alpha}{\sin \alpha} \sin \beta + \dfrac{\beta}{\sin \beta} \sin \gamma + \dfrac{\gamma}{\sin \gamma} \sin \alpha \leq \dfrac{\alpha}{\sin \alpha} \sin \alpha + \dfrac{\beta}{\sin \beta} \sin \beta + \dfrac{\gamma}{\sin \gamma} \sin \gamma = \alpha + \beta + \gamma.$
