What is the maximum value of f''(0) for functions in set F?

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The maximum value of f''(0) for functions in the set F, defined as F={f:D→D | ∀z∈D ∂̅zf=0}, is calculated using Cauchy's estimates. The supremum L=|f''(0)| can be achieved by a specific function g in F, demonstrating that g''(0)=L. The discussion emphasizes the importance of the maximum modulus principle and Cauchy estimates in determining the behavior of holomorphic functions within the unit disk.

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Let D\subset\mathbb{C} be the unitdisc and F=\{f:D\rightarrow D\,|\,\forall z\in D\partial_{\bar{z}}f=0\}, calculate L=\sup_{f\in F}|f''(0)|. Show that there is an g\in F with g''(0)=L.
I am a bit stuck. But I think that it might be an idea to start with Cauchy estimate. Any other ideas?
 
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<br /> |a_n|\leq\frac{1}{2\pi}\frac{M}{r^3}l=\frac{M}{r^2}\quad M=\max_{|z|&lt;r&lt;1}|f(z)|=\sup_{z\in\partial D_r}|f(z)|<br />
?
 
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=1??
 

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