Let $ \ \large f_k \ $ are harmonic on a Unit Disc $ \ D \ $

Question

Let $ \ \large f_k \ $ are harmonic on a Unit Disc $ \ D \ $. Show that no linear combination of these $ \large \ f_k \ $ can be Negative on $ \ \partial D \ $ and positive at some point in the Interior of $ \ D \ $.

( Hint: Cauchy Integral formula, Maximum-minimum principle )

Answer:

Given $ f_k \ $ are harmonic .

By Maximum - Minimum Principle , $ \ f_k \ $ attains its maximum on Boundary $ \partial D \ $.

Thus $ f_k \ $ can not be positive at Interior of $ D \ $ and Negative value on boundary $ \partial D \ $.

Thus the linear combination of these $ f_k \ $ also can not assume positive value in the interior of $ D \ $ and Negative value at the boundary $ \partial D \ $.

Am I right ?

Is there any help?

Answer 1

Easy fact Let $$F =\sum a_kf_k$$ be any finite linear combination of $f_k$ then is easy to see that $F$ is Harmonic too.

Therefore it archives its maximum and its minimum on $\partial D$

So if $$F\ge 0 ~~\text{on}~\implies \min_D F\ge \min_{\partial D}F\ge0 $$

Hence $F$ cannot be Negative on ∂D and positive at some point in the Interior of D

Therefore such linear combination does not exists