I came across the following assertion and am having trouble justifying it:
If $z$ is a nonzero complex number with $|z| \leq \pi/2$ and $|\sin z| \leq 1/4$, then $ \left| \frac{z}{\sin z} \right| \leq \frac{1/4}{\sin(1/4)} = 1.0104931\ldots $
I would appreciate some help. Thanks.
EDIT: Andrew has pointed out that the above inequality fails if $z = \sin^{-1}(1/4) = 0.25268025\ldots$. After more thought, I figured out that if $z$ is real with $|z| \leq \pi/2$ and $|\sin z| \leq 1/4$, then $|z| \leq \sin^{-1}(1/4)$ and $z / \sin z$ is increasing in $[0,\sin^{-1}(1/4)]$; consequently, $ \left| \frac{z}{\sin z} \right| \leq \frac{\sin^{-1}(1/4)}{1/4} = 1.0107210\ldots $ holds.