# Real fixed points and singular values of family of functions arising from generating function of unified generalized Apostol-type polynomials

Volume 12, Issue 9, pp 602--610 Publication Date: May 06, 2019       Article History
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### Authors

Mohammad Sajid - College of Engineering, Qassim University, Buraidah, Al-Qassim, Saudi Arabia.

### Abstract

Our main objective is to study the real fixed points and singular values of a two-parameter family of transcendental meromorphic functions $g_{\lambda,n}(z)=\lambda \frac{z}{(b^{z}-1)^{n}}$, $\lambda \in \mathbb{R} \backslash \{0\}$, $z \in \mathbb{C} \backslash \{0\}$, $n\in \mathbb{N} \backslash \{1\}$, $b>0$, $b\neq 1$ in the present paper which obtains from generating function of the unified generalized Apostol-type polynomials. The real fixed points of $g_{\lambda,n}(x)$, $x\in {\mathbb{R}}\setminus \{0\}$ with their stability are found for $n$ odd and $n$ even. It is shown that $g_{\lambda,n}(z)$ has infinite number of singular values. Further, it is seen that some critical values of $g_{\lambda,n}(z)$ lie in the closure of the disk and other lie in the exterior of the disk with center at the origin.

### Keywords

• Real fixed points
• critical values
• singular values
• meromorphic function

•  30D30
•  37C25
•  58K05

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