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?