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 http://dx.doi.org/10.22436/jnsa.012.09.05

Publication Date: May 06, 2019 Submission Date: January 13, 2019 Revision Date: April 03, 2019 Accteptance Date: April 10, 2019

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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.

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Keywords

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

MSC

•  30D30
•  37C25
•  58K05

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