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

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In summary, "L" represents the limit of the second derivative of a function "f" at the point 0, where "f" is a member of a given set. The limit L is calculated by finding the supremum of all the second derivatives of the functions in the set at the point 0. "sup" stands for supremum, which is the least upper bound of a set, and in this case, it represents the maximum value of the second derivatives of the functions in the set at the point 0. The limit L is related to the concavity of a function as it represents the maximum rate of change of the slope at a specific point. The equation L=sup{f''(0)|f in
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niklas
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Let [tex]D\subset\mathbb{C}[/tex] be the unitdisc and [tex]F=\{f:D\rightarrow D\,|\,\forall z\in D\partial_{\bar{z}}f=0\}[/tex], calculate [tex]L=\sup_{f\in F}|f''(0)|[/tex]. Show that there is an [tex]g\in F[/tex] with [tex]g''(0)=L[/tex].
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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[tex]
|a_n|\leq\frac{1}{2\pi}\frac{M}{r^3}l=\frac{M}{r^2}\quad M=\max_{|z|<r<1}|f(z)|=\sup_{z\in\partial D_r}|f(z)|
[/tex]
?
 
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=1??
 

1. What does "L" represent in the equation L=sup{f''(0)|f in the set }?

"L" represents the limit of the second derivative of the function "f" at the point 0, where the function "f" is a member of the given set.

2. How is the limit L calculated for a given set of functions?

The limit L is calculated by finding the supremum (or least upper bound) of all the second derivatives of the functions in the given set at the point 0.

3. What does "sup" mean in the equation L=sup{f''(0)|f in the set }?

"sup" stands for supremum, which is the least upper bound of a set. In this case, it represents the maximum value of all the second derivatives of the functions in the given set at the point 0.

4. How is the limit L related to the concavity of a function?

The limit L represents the maximum rate of change of the slope of a function at a specific point. So, a higher value of L indicates a sharper curve or a more concave function at that point.

5. What does the equation L=sup{f''(0)|f in the set } tell us about the behavior of the functions in the given set?

This equation tells us about the maximum possible curvature of the functions in the given set at the point 0. It helps us understand the overall shape and behavior of the functions in the set near the point 0.

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