If \(\varphi:[-\delta,\delta]\to [0,\infty)\), \(\delta>0\), is a convex function with \(\varphi(0)=0\), a subharmonic saddle for \(\varphi\) is a subharmonic function \(u\) on \(|z|\leq \delta'\), \(0 -\infty. \] The theorem is proved by first proving a result concerning plurisubharmonic saddles for several complex variables, and then using a theorem of Warschawski and Tsuji on the existence of angular derivatives of conformal mappings.