Basic Complex Analysis: Uniform convergence of derivatives to 0

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SUMMARY

The discussion focuses on the uniform convergence of the derivatives of a sequence of holomorphic functions, specifically proving that if the sequence \( f_n \) converges uniformly to zero in the disc \( D_1 = \{ z : |z| < 1 \} \), then the derivatives \( f'_n \) converge uniformly to zero in the smaller disc \( D = \{ z : |z| < 1/2 \} \). Utilizing the Cauchy inequalities derived from the Cauchy integral formula, it is established that for any \( \varepsilon > 0 \), there exists an \( N \) such that for \( n > N \), the sup norm of \( f'_n(z) \) is less than \( 2\varepsilon \), confirming the desired convergence.

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Homework Statement


Let f_n be a sequence of holomorphic functions such that f_n converges to zero uniformly in the disc D1 = {z : |z| < 1}. Prove that f &#039;_n converges to zero uniformly in D = {z : |z| < 1/2}.

Homework Equations


Cauchy inequalities (estimates from the Cauchy integral formula)

The Attempt at a Solution


Okay I am less sure about this one, but

Given \varepsilon &gt; 0, there exists N > 0 such that n > N implies
||f_n|| &lt; \varepsilon.
Fix 0 < r < 1/2. Then for any z in D, the Cauchy inequalities imply
f&#039;_n(z) \leq \frac{\sup_{z \in D} |f_n (z)|}{r} &lt; 2\varepsilon,
whence we have sup norm convergence of the f&#039;_n to 0, as desired.
 
Last edited:
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That looks pretty straightforward to me. I don't see anything wrong with it.
 

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