if A,B,C are acute then is it true : cyclic sum ( ((sin x)/(sin y)) le ((x)/(y)) le ((tan x)/(tan y)) )? If $A,B,C$ are acute angles then is it always true :
$$\sum_{cyc} \frac{ \sin A}{ \sin B} \le \sum_{cyc} \frac{A}{B} \le \sum_{cyc} \frac{ \tan A}{ \tan B}$$ 
where
$\sum_{cyc}$ denotes the cyclic sum.
 A: Yes. This follows from the following 
Lemma. If $x,y,z>0$ and $xyz=XYZ=1$ and $\min(x,y,z)\leqslant \min(X,Y,Z)$, $\max(x,y,z)\geqslant \max(X,Y,Z)$, then $x + y + z\geqslant X + Y + Z$.
Proof. Move maximal and minimal elements in $\{x,y,z\}$ making them closer to each other with fixed product. The sum decreases and we reach the situation when minimal or maximal elements of $\{x,y,z\}$ and $\{X,Y,Z\}$ are equal. After that move two other elements of $\{x,y,z\}$.
Actually the lemma is a partial case of Karamata's inequality.
Now note that if $\pi/2>A>B>0$, then $\sin A/\sin B<A/B<\tan A/\tan B$, as $\frac{\sin x}x$ decreases and $\frac{\tan x}x$ increases on $(0,\pi/2)$. The last property follows from convexity of $\tan$ and concavity of $\sin$.
Let $\{x,y,z\}=\{A/B,B/C,C/A\}$, $\{X,Y,Z\}=\{\sin A/\sin B,\sin B/\sin C,\sin C/\sin A\}$. Without loss of generality, the maximal element of $\{X,Y,Z\}$ is $\sin A/\sin B$ where $A\geqslant B$, then $\{x,y,z\}$ contains the element $A/B\geqslant \sin A/\sin B$. Analogously for minimal elements and for tangents.
