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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

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

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