Let $ f $ be a real-valued $ \operatorname{BMO} $ function on $ \mathbb{R}^n $. Prove that the functions $$ f_{KL}(x)=\left\{\begin{aligned} &K&\text{ if }&f(x)<K,\\ &f(x)&\text{ if }&K\leq f(x)\leq L,\\ &L&\text{ if }&f(x)>L, \end{aligned}\right. $$ satisfy $ \|f_{KL}\|_{\operatorname{BMO}}\leq\frac{9}{4}\|f\|_{\operatorname{BMO}} $.
If the constant $ \frac{9}{4} $ is replaced by $ 3 $, I know how to show this result. Can you give me some hints or references?
