Singular values of three-parameter families $\lambda (\frac{e^{az}-1}{z})^{\mu}$ and $\lambda (\frac{z}{e^{az}-1})^{\eta}$

Abstract

The main goal of the present paper is to investigate the singular values of three-parameter families of transcendental (i) entire functions $f_{\lambda,a,\mu}(z)=\lambda\bigg(\dfrac{e^{az}-1}{z}\bigg)^{\mu}$ and $\;f_{\lambda,a,\mu}(0)=\lambda a^{\mu}$; $\; \mu> 0$, $\; \lambda, \; a\in \mathbb{R} \backslash \{0\}$, $\; z \in \mathbb{C}\;$ (ii) meromorphic functions $g_{\lambda,a,\eta}(z)=\lambda\bigg(\dfrac{z}{e^{az}-1}\bigg)^{\eta}$ and $\;g_{\lambda,a,\eta}(0)=\frac{\lambda}{a^{\eta}}$, $\eta> 0$; $\lambda, a\in \mathbb{R} \backslash \{0\}$, $z \in \hat{\mathbb{C}}$. It is obtained that all the critical values of $f_{\lambda,a,\mu}(z)$ and $g_{\lambda,a,\eta}(z)$ lie in the right half plane for $a<0$ and in the left half plane for $a>0$. It is also shown that all these critical values of $f_{\lambda,a,\mu}(z)$ and $g_{\lambda,a,\eta}(z)$ are interior and exterior of the open disk centered at origin and having radii $|\lambda a^{\mu}|$ and $|\frac{\lambda}{a^{\eta}}|$ respectively. Further, it is described that the functions $f_{\lambda,a,\mu}(z)$ and $g_{\lambda,a,\eta}(z)$ both have infinitely many singular values.