‎We show that‎, ‎in complex interpolation‎, ‎an operator function that is compact‎ ‎on one side of the interpolation scale will be compact for all proper‎ ‎interpolating‎ ‎spaces if the right hand side $(Y^0,Y^1)$ is reduced to a single space‎. ‎A corresponding result‎, ‎in restricted generality‎, ‎is shown if the left hand side‎ ‎$(X^0,X^1)$ is reduced to a single space‎. ‎These results are derived from the fact that a holomorphic operator valued‎ ‎function on an open subset of $\mathbb{C}$ which‎ ‎takes values in the compact operators on part of the boundary is in fact compact‎ ‎operator valued‎.