This is taken from Qin Han's book Nonlinear elliptic equations of the second order.
(Theorem 2.2.1) Let $a_{ij} \in C^1(B_R \times \mathbb{R}^n)$ satisfy the uniform ellipticity and for any $x \in B_R$ and $p \in \mathbb{R}^n$ there are some positive constants $A_0,A_1$ such that $$R | \nabla_x a_{ij}(x,p)| \leq A_0, \qquad |p| |\nabla_p a_{ij}(x,p)| \leq A_1 $$ Suppose $u \in L^\infty(B_R) \cap C^3(B_R)$ is a solution of $$a_{ij}(x,\nabla u(x)) = f(x) \quad \text{ in } B_R $$ for some $f \in C^1(B_R)$ with $|f|_{C^1(B_R)} < \infty$ Then $$R |\nabla u|_{L^\infty(B_{R/2})} \leq C (|u|_{L^\infty(B_R)} + R^2 |f|_{L^\infty(B_R)} + R^3 |\nabla f|_{L^\infty(B_R)}) $$
I understand the proof except the last part. He proves that $$\sup_{B_R} (\eta^2 |\nabla u|^2 + \frac{c_1}{\lambda R^2} u^2 ) \leq C \sup_{B_R} (\frac{u^2}{R^2} + R^2 f^2 +R^4 |\nabla f|^2) $$
where $\eta \in C_0^2(B_R)$ is a cut off function with $0 \leq \eta \leq 1$, $\eta = 1$ in $B_{R/2}$ and $$ |\nabla \eta|^2+|\nabla^2 \eta| \leq c_0$$ and $\lambda$ is the constant lower bound of the uniform ellipticity.
I dont understand how that estimate implies the first one, I managed to prove that $$|\nabla u|_{L^\infty(B_{R/2})}^2 \leq C(\frac{|u|_{L^\infty(B_R)}^2}{R^2} + R^2 |f^2|_{L^\infty(B_R)} + R^4 |\nabla f|^2_{L^\infty(B_R)}) $$ but I dont see how that implies the desired estimate. Any help appreciated
Pd: Han defines the $L^\infty$ norm as $|u|_{L^\infty(\Omega)} = \sup_{\Omega} |u|$
Take the square root on both sides and then multiply both sides by $R$. Then on the right hand side use the inequality
$$\sqrt{a+b+c} \leq \sqrt{a} + \sqrt{b} + \sqrt{c},$$
for nonnegative numbers $a,b,c$ (to see this, just square both sides).