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Let $B$ be an open bounded subset of $C$, and $ dB$ denotes the boundary of $B$.

Can we say, for every entire function $f$, we have $df(B) = f(dB)$ ? If no, under what condition would it hold?

If I use open map theorem, we see that interior point would go to interior point. But there won't be any boundary point ( because boundary point should be a member of the set, though there could be frontier points). So I guess $f(dB)$ would be empty. So it would be equal to $df(B)$ of image set is also open?

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Hint: Let $B$ be the upper half of the open unit disc. Define $f(z) = z^3.$ What are $f(\partial B), \partial (f(B))?$

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