  We show that for a real transcendental meromorphic function f, the
differential polynomial f'+f^m with m > 4 has infinitely many non-real zeros.
Similar results are obtained for differential polynomials f'f^m-1. We specially
investigate the case of meromorphic functions with finitely many poles. We show
by examples the precision of our results. One of our main tools is the Fatou
theorem from complex dynamics.
