Abstract

The singular values of two parameter families of entire functions $f_{\lambda,a}(z)=\lambda\frac{e^{az}-1}{z}$, $f_{\lambda,a}(0)=\lambda a$ and meromorphic functions $g_{\lambda,a}(z)=\lambda\frac{z}{e^{az}-1}$, $g_{\lambda,a}(0)=\frac{\lambda}{a}$, $\lambda, a \in \mathbb{R} \backslash \{0\}$, $z \in \mathbb{C}$, are investigated. It is shown that all the critical values of $f_{\lambda,a}(z)$ and $g_{\lambda,a}(z)$ lie in the right half plane for $a<0$ and lie in the left half plane for $a>0$. It is described that the functions $f_{\lambda,a}(z)$ and $g_{\lambda,a}(z)$ have infinitely many singular values. It is also found that all the singular values $f_{\lambda,a}(z)$ are bounded and lie inside the open disk centered at origin and having radius $|\lambda a|$ and all the critical values of $g_{\lambda,a}(z)$ belong to the exterior of the disk centered at origin and having radius $|\frac{\lambda}{a}|$.

This work is licensed under a Creative Commons Attribution 4.0 License.