What is the Proof for Locally Uniformly Convergence of Series in the Unit Disc?

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In summary, we are trying to prove that the series Ʃf(z^n) is locally uniformly convergent in the unit disc B(0,1), given that f(z) is holomorphic in B(0,1) and f(0)=0. We can use the Cauchy formula to find the bounds on the derivatives and then apply the Weierstrass M-test to show uniform convergence in B(z,r). However, we still need to prove uniform convergence in a neighborhood of each point in B(0,1).
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Homework Statement


Let f(z) be holomorphic in the unit disc B(0,1),such thaf f(0)=0.Prove that the series Ʃf(z^n) is locally uniformly convergence in B(0,1).

Homework Equations


locally uniformly convergence:if it is uniformly convergence in a neibourhood of each point of B(0,1).

The Attempt at a Solution


At each point z in the disc there is an r such that the closure of B(z,r) is contained in B(0,1).I tried to use the cauchy formula for finding bounds on the derivatives so that the required series will be uniformly convergence in B(z,r),from Weierstrass M-test.I proved that it uniformly converges in B(0,r) for certain r.
 
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But,i can't prove that it will be uniformly convergence in a neibourhood of each point of B(0,1).Please help me.
 

Related to What is the Proof for Locally Uniformly Convergence of Series in the Unit Disc?

1. What is locally uniformly convergence?

Locally uniformly convergence is a mathematical concept that describes the behavior of a sequence of functions. It means that the functions in the sequence converge uniformly on every compact subset of the domain.

2. How is locally uniformly convergence different from pointwise convergence?

Pointwise convergence only requires that the functions in the sequence converge at each individual point in the domain. Locally uniformly convergence, on the other hand, requires the functions to converge uniformly on every compact subset of the domain.

3. What is the significance of locally uniformly convergence in analysis?

Locally uniformly convergence is an important concept in analysis because it allows us to extend properties and theorems from the individual functions in a sequence to the limit function. This is useful in many areas of mathematics, including differential equations and Fourier analysis.

4. Can a sequence of functions be locally uniformly convergent but not uniformly convergent?

Yes, it is possible for a sequence of functions to be locally uniformly convergent but not uniformly convergent. This means that the functions in the sequence converge uniformly on every compact subset of the domain, but there may be points in the domain where the convergence is not uniform.

5. How is locally uniformly convergence related to continuity?

Locally uniformly convergence is closely related to continuity because it implies that the limit function is continuous. This is because a sequence of continuous functions that converges locally uniformly will have a continuous limit function. However, the converse is not necessarily true - a continuous limit function does not always imply locally uniformly convergence.

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