Holomorphic bounded by log

Let $f$ be a holomorphic function in $\{z\Big| 0<|z|<1\}$, continuous in $\{z\Big| 0<|z|\le 1\}$ and $|f(z)|\le \log\left(\frac{1}{|z|}\right)$. Prove that $f\equiv 0$.
Proof. Since $f$ is continuous on the boundary, for $z_0$, s.t. $|z_0|=1$ we have $$|f(z_0)|=\lim_{z\to z_0}|f(z)|\le \lim_{z\to z_0}\log\left(\frac{1}{|z|}\right)=0$$ so $f$ is $0$ on the boundary. Consider now $g(z)=zf(z)$. So $g$ is also $0$ on the boundary. Now observe that $$\lim_{z\to 0}|g(z)|=\lim_{z\to 0}|z||f(z)|\le \lim_{z\to 0}|z|\log\left(\frac{1}{|z|}\right)=0$$ so $0$ is removable singularity for $g$. If we define $g(0)=0$, then $g$ is holomorphic on $\{z\Big||z|<1\}$, continuous on the boundary. From the maximum modulus principle $|g|$ attains its maximum at the boundary, and it is $0$ there, so $g\equiv 0$ on $\{z\Big||z|<1\}$. Now take $z_0\ne 0$, and then $f(z_0)=\frac{g(z_0)}{z_0}=0$, so $f\equiv 0$.

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