On the zeros of $L’+L^ 2$ for certain rational functions $L$

Abstract:

Let $L$ be a nonconstant rational function whose poles are real, simple with each one having a positive residue. Then, if $L’ + {L^2}$ has no nonreal zeros, $L$ has the form \[ L(z) = \sum \limits _{k = 1}^n {\frac {{{\alpha _k}}}{{z - {x_k}}} - az + b,} \] ${x_k}$ are real, ${\alpha _k} > 0$ for $1 \leq k \leq n,a \geq 0$ and $b$ is real. In particular, if $P$ is a polynomial of degree $\geq 2$, then $P’ + {P^2}$ has nonreal zeros. The result is applied to entire functions in connection with zeros of the derivatives.