Singular Values of One Parameter Family \(\lambda ((e^{z}-1)/z)^{m}\)






Critical values, Singular values


In the present paper, the singular values of one parameter family of entire functions \(f_{\lambda}(z)=\lambda\bigg(\dfrac{e^{z}-1}{z}\bigg)^{m}\) and \(f_{\lambda}(0)=\lambda\), \(m\in \mathbb{N}\backslash \{0\}\), \(\lambda\in \mathbb{R} \backslash \{0\}\), \(z \in \mathbb{C}\) is investigated. It is shown that all the critical values of \(f_{\lambda}(z)\) lie in the left half plane. It is also found that the function \(f_{\lambda}(z)\) has infinitely many bounded singular values and lie inside the open disk centered at origin and having radius \(|\lambda|\).

Author Biography

Mohammad Sajid, Qassim University

College of Engineering,

Assistant Professor


[1] R. L. Devaney, Complex dynamics and entire functions. Complex Dynamical Systems: The Mathematics behind the Mandelbrot and Julia sets, Proc. Symp. Appl. Math., Ed. R. L. Devaney. Amer. Math. Soc., Vol. 49, (1994), pp.181-206.

[2] R. L. Devaney, Sex: Dynamics, topology, and bifurcations of complex exponentials, Topology and its Applications, 110 (2001) 133--161. doi:10.1016/S0166-8641(00)00099-7

[3] T. Kuroda, C. M. Jang, Julia set of the Function z exp(z+ µ) II, Tohoku Math. Journal, 49 (1997) 557--584. doi:10.2748/tmj/1178225063

[4] T. Nayak, M. G. P. Prasad, Julia Sets of Joukowski-Exponential Maps, Complex Anal. Oper. Theory, 8 (2014) 1061--1076. doi: 10.1007/s11785-013-0335-1

[5] P. Petek, M. S. Rugelj, The Dynamics of λ + z + exp(z), J. Math. Anal. Appl., 222 (1998) 38--63. doi:10.1006/jmaa.1997.5724

[6] M. G. P. Prasad, T. Nayak, Dynamics of certain class of critically bounded entire transcendental functions, J. Math. Anal. Appl., 329 (2007) 1446--1459. doi:10.1016/j.jmaa.2006.06.095

[7] M. Sajid, Singular Values and Fixed Points of Family of Function zez/(ez-1), International Journal of Applied Mathematics, 27 (2014) 147--154. doi:10.12732/ijam.v27i2.4

[8] M. Sajid, Singular Values of a Family of Singular Perturbed Exponential Map, British Journal of Mathematics and Computer Science, 4 (2014) 1678--1681. doi:10.9734/BJMCS/2014/9598

[9] M. Sajid, Singular Values and Fixed Points of Family of Generating Function of Bernoulli's Numbers, J. Nonlinear Sci. Appl., 8 (2015) 17--22.

[10] G. P. Kapoor, M. G. P. Prasad, Dynamics of (ez-1)/z: the Julia set and Bifurcation, Ergod. Th. & Dynam. Sys., 18 (1998) 1363--1383.

[11] W. Bergweiler, M. Haruta, H. Kriete, H. G. Meier, N. Terglane, On the limit functions of iterates in wandering domains, Ann. Acad. Sci. Fenn. Series A. I. Math, 18 (1993) 369--375.

[12] S. Morosawa, Y. Nishimura, M. Taniguchi, T. Ueda, Holomorphic Dynamics, Cambridge University Press, Cambridge, 2000.

[13] J.H. Zheng, On fixed-points and singular values of transcendental meromorphic functions, Science China Mathematics, 53 (2010) 887--894. doi: 10.1007/s11425-010-0036-4

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