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 $\mathrm{\Delta }$ such that: (i) at least one of $f$ and $1\mathrm{/}f$ has finitely many poles and non-real zeros in $\mathrm{\Delta }$; (ii)~$f^{(k)}$ has finitely many non-real zeros in $\mathrm{\Delta }$, for some $k\ge 2$.