Hölder estimates of derivatives of harmonic function?

Question

In Gilbarg and Trudinger's textbook about elliptic equation, they give the following interior estimate of derivatives of harmonic function $v$ in $\Omega\subset \mathbb{R}^n$ using the mean value theorem: for any $\Omega'\Subset \Omega$, denote $d=dist(\Omega',\partial \Omega )$, then we have $$\sup_{\Omega'} |\partial^k u |\le \frac{C}{d^k}\sup_{\Omega}|u|$$ where $C$ depends on $n,k$. Moreover, I want to establish a interior Hölder estimate for $u$: for $B_1=B_R(x), B_2=B_{2R}(x)\subset \Omega$, $$|u|_{2,\alpha;B_1}'\le C|u|_{0;B_2}, \text{ i.e. } R|\partial u |_{0;B_1}+R^2|\partial^2 u |_{0;B_1}+R^{2+\alpha}[\partial^2 u ]_{0;B_1}\le C|u|_{0;B_2} .$$ The first two terms can be controlled using the inequality by G-T directly, but how about the last term involving Hölder seminorm? I attempt to apply the mean value property again: \begin{equation} \begin{split} \frac{\partial^2 u (x)-\partial^2 u(y) }{|x-y|^\alpha} =\frac{1}{\omega_nR^n} \frac{ \int_{\partial B_R(x)} -\int_{\partial B_R(y) }\partial u(s)ds }{|x-y|^\alpha}. \end{split} \end{equation} However, the right hand sees hard to estimate as it involves the difference between integral area. Can anyone give a more efficient method to estimate the Hölder term?