Derivatives of Meromorphic Functions with Multiple Zeros and Small Functions

Abstract

Let $f\left(z\right)$ be a meromorphic function in $ℂ$, and let $\alpha \left(z\right)=R\left(z\right)h\left(z\right)\not\equiv 0$, where $h\left(z\right)$ is a nonconstant elliptic function and $R\left(z\right)$ is a rational function. Suppose that all zeros of $f\left(z\right)$ are multiple except finitely many and $T\left(r,\alpha \right)=o\left\{T\left(r,f\right)\right\}$ as $r\to \infty$. Then $f^{\text{'}}\left(z\right)=\alpha \left(z\right)$ has infinitely many solutions.