A note on Hayman’s problem and the sharing value

Abstract

Let $f$ be a nonconstant meromorphic functions, $n, k$ be two positive integers. Suppose that $f^{n}$ and $(f^{n})^{(k)}$ share the value $a( \ne 0,\infty )$ CM. If either (1) $n>k+2$, or (2) $n>k+1$ and $\bar{N}(r,f)=\lambda T(r,f)(\lambda \in [0,\frac{1}{2}))$, then $f^{n}=(f^{n})^{(k)}$ and $f$ assumes the form \begin{equation*} f(\mathrm{z}) = \mathrm{ce}^{\frac{λ}{n}z} \end{equation*} where $c$ is a nonzero constant and ${\lambda}^{k}=1$.