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

Mohammad Sajid


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.

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Mohammad Sajid, Singular values of three-parameter families $\lambda (\frac{e^{az}-1}{z})^{\mu}$ and $\lambda (\frac{z}{e^{az}-1})^{\eta}$, Journal of Mathematical and Computational Science, Vol 8, No 2 (2018), 270-277

Copyright © 2018 Mohammad Sajid. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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