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

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 )

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?

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