Certain dynamical aspects of a family $f_\lambda(z)= \lambda \frac{e^{z}}{z+1}\ \mbox{for}\ z\in \mathbb{C}$ when $\lambda < 0$

Abstract

We study the change of dynamics of transcendental meromorphic functions
$f_\lambda= \lambda \frac{e^{z}}{z+1}\ \mbox{for}\ z\in \mathbb{C}$ when $\lambda $ varies on the negative real axis. It is shown that there is a $\widehat{\lambda}$ such that the Fatou set of $f_\lambda$ is empty for $ \lambda < \widehat{\lambda}$ whereas the
Fatou set is an invariant parabolic basin
corresponding to a real rationally indifferent fixed point $ \widehat{x}$ if $\lambda=\widehat{\lambda}$ . In fact, the Fatou set is an invariant attracting basin of a real
negative fixed point $\widehat{a}_\lambda $ if $\widehat{\lambda} < \lambda < 0$. Also the dynamics of $f_\lambda^n$ for $n\geq 2$ at the fixed points is investigated for different values of $\lambda$. As a generalization of $f_\lambda$, we observed some dynamical issues for the class of entire maps $F_{\lambda,a,m}(z)=\lambda(z+a)^m\exp(z)$ where $\lambda, a \in \mathbb{C}$ and $m \in \mathbb{N}$.