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Consider a simply connected domain bounded by a Jordan curve, and cut out two regions inside the domain, also bounded by Jordan curves. If $f(z)$ is a holomorphic function on the domain with bounds $B_1$, $B_2$, and $B_3$ on the boundary curves, how can we obtain a good bound (better than just the maximum of the $B_i$) on $f(z)$ inside the domain?

In the doubly-connected case, perhaps conformal mapping and the three lines lemma could be used, but I don't see how to proceed here.

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We can bound $\log |f(z)|$ using the maximum principle for subharmonic functions. We can construct harmonic measures corresponding to each of the boundary curves and take a linear combination with coefficients $B_1$, $B_2$, $B_3$ to get a pointwise bound inside the domain.

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