Non-real zeros of derivatives in the unit disc

The main result establishes an estimate for the growth of a real meromorphic function f on the unit disc Δ such that: (i) at least one of f and $1/f$ has finitely many poles and non-real zeros in Δ; (ii)~f⁽ᵏ⁾ has finitely many non-real zeros in Δ, for some k ≥ 2.