Actually this question is derived from the Rudin's Real and Complex Analysis's lemma 15.17.
In the lemma, given condition is below; Let $h(z) \in H(\Omega)$ such that Re $h(z) = \log |1-z|$, |Im $h(z)| < \frac{\pi}{2} $, and take $\gamma(t)$ be path from $e^{i\delta}$ to $e^{-i\delta}$ on $\partial D(1;|1-e^{i\delta}|)$. Then Rudin says that
|Re$[\frac{1}{2*\pi}\int_{\gamma}\frac{h(z)dz}{z}]| < C \delta \log(1/\delta)$
It also says that a length of $\gamma$ is less than $\pi \delta$. I know he uses M-L inequality, but do not now how to approximate $|h(z)/z|$ on $z=\gamma(t)$ to $\log(1/\delta)$. Could you give some hint why such approximation makes sense?