In other words, what is the value of the maximum mean oscillation of the sine function over an interval, \begin{align} || \sin ||_{\mathrm{BMO}} &= \sup_{a,b\in\mathbb{R}} \frac{1}{|b-a|}\int_a^b |\sin(x) - \frac{1}{|b-a|}\int_a^b \sin(y) dy | \:dx \\ &= \sup_{a<b} \frac{1}{(b-a)^2}\int_a^b |\sin(x) - cos(a)+cos(b) | \:dx\; ? \end{align} The BMO-norm is nice to work with in proofs, but I don't know the value of it in any non-trivial (non-constant) example. This makes it hard for me to understand (get an intuition for) what it really means when the BMO-norm of a function is "small" or "large".
I don't know how to deal with the absolute value in the integral, maybe one could also estimate it numerically? But should it not be something simple? Thanks for your help.