Meromorphic functions and their k-th derivative share only one small function CM

Amer H. H. Al-Khaladi

It is shown that if a non-constant meromorphic function $f$ and its derivative $f^{(k)}$ share one meromorphic small function $\beta\not\equiv 0,\infty$ CM (counting multiplicities), then either $T(r,f^{(k)})=O(\bar N(r,\frac{1}{f^{(k)}}))$ or $f-\beta=(1-\frac{p_{k-1}}{\beta})(f^{(k)}-\beta)$, where $p_{k-1}$ is a polynomial of degree at most $k-1$ and $1-\frac{p_{k-1}}{\beta}\not\equiv 0$. This result answers Brück's question and improves Al-Khaladi's result.