How to show following inequality? Suppose we have
\begin{align*}M(r)-m(2 r) &\leq C(m(r)-m(2 r))\\
M(2 r)-m(r) &\leq C(M(2 r)-M(r)). \end{align*}
From that how to show that $$
\omega(r) \leq \frac{C-1}{C+1} \omega(2 r)
$$
where $\omega(r)=M(r)-m(r)$  and $m(r)=\underset{B_{r}}{\operatorname{essinf}} u, \quad M(r)=\operatorname{esssup}_{B_{r}} u$
My attempt:
\begin{align*}
M(r)-m(2 r) &\leq C(m(r)-m(2 r))\\
\implies M(r)-m(r)&\le (C-1)(m(r)-m(2r))\\
\omega(r)&\le (C-1)(m(r)-m(2r))
 \end{align*}
Also
\begin{align*}
M(2 r)-m(r) &\leq C(M(2 r)-M(r))\\
\implies M(r)-m(r)&\le (C-1)(M(2r)-M(r))\\
\omega(r)&\le (C-1)(M(2r)-M(r))
 \end{align*}
From that how to show that  $$
\omega(r) \leq \frac{C-1}{C+1} \omega(2 r)
$$
Any help will be appreciated

Add the two inequalities you started with. You get
$$ \omega(2r) + \omega(r) \leq C (\omega(2r)- \omega(r)) $$
Now just rearrange.
