Parametrization of entire functions of sine-type by their critical values

1992 ◽  
pp. 237-242 ◽  
Author(s):  
A. Eremenko ◽  
M. Sodin
Keyword(s):  
Entire Functions ◽  
Critical Values

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

2015 ◽  
Vol 4 (2) ◽  
pp. 295 ◽  
Author(s):  
Mohammad Sajid
Keyword(s):  
Half Plane ◽  
Entire Functions ◽  
Singular Values ◽  
Critical Values ◽  
Parameter Family ◽  
Open Disk ◽  
Left Half ◽  
Left Half Plane

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 $|\lambda|$.

scholarly journals Singular Values of Two Parameter Families $\lambda\dfrac{e^{az}-1}{z}$ and $\lambda\dfrac{z}{e^{az}-1}$

2016 ◽  
Vol 8 (1) ◽  
pp. 10 ◽  
Author(s):  
Mohammad Sajid
Keyword(s):  
Half Plane ◽  
Meromorphic Functions ◽  
Entire Functions ◽  
Singular Values ◽  
Critical Values ◽  
Open Disk ◽  
Left Half ◽  
Left Half Plane ◽  
The Right ◽  
Two Parameter

<p>The singular values of two parameter families of entire functions $f_{\lambda,a}(z)=\lambda\frac{e^{az}-1}{z}$, $f_{\lambda,a}(0)=\lambda a$ and meromorphic functions $g_{\lambda,a}(z)=\lambda\frac{z}{e^{az}-1}$, $g_{\lambda,a}(0)=\frac{\lambda}{a}$, $\lambda, a \in \mathbb{R} \backslash \{0\}$, $z \in \mathbb{C}$, are investigated. It is shown that all the critical values of $f_{\lambda,a}(z)$ and $g_{\lambda,a}(z)$ lie in the right half plane for $a&lt;0$ and lie in the left half plane for $a&gt;0$. It is described that the functions $f_{\lambda,a}(z)$ and $g_{\lambda,a}(z)$ have infinitely many singular values. It is also found that all the singular values $f_{\lambda,a}(z)$ are bounded and lie inside the open disk centered at origin and having radius $|\lambda a|$ and all the critical values of $g_{\lambda,a}(z)$ belong to the exterior of the disk centered at origin and having radius $|\frac{\lambda}{a}|$.</p>

Conformal measures for non-entire functions with critical values eventually mapped onto infinity

Analysis ◽  
2005 ◽  
Vol 25 (4) ◽  
Author(s):  
Janina Kotus
Keyword(s):  
Meromorphic Functions ◽  
Fractal Structure ◽  
Entire Functions ◽  
Critical Values ◽  
Conformal Measures

SummaryThis paper is a continuation of our earlier works [7] and [9] on the fractal structure of expanding and subexpanding meromorphic functions of the form

Entire functions with Julia sets of zero measure

1990 ◽  
Vol 108 (3) ◽  
pp. 551-557 ◽  
Author(s):  
Gwyneth M. Stallard
Keyword(s):  
Entire Functions ◽  
Measure Zero ◽  
Julia Set ◽  
Julia Sets ◽  
Critical Values ◽  
Self Similarity ◽  
Zero Measure ◽  
Plane Measure

AbstractWe extend results of McMullen about the dynamics of entire functions for which the orbits of the critical values stay away from the Julia set. In particular we show that such functions are expanding on their Julia sets which have self-similarity properties. Under suitable further conditions the Julia sets have plane measure zero.

Critical Values

2008 ◽  
Vol 3 (3) ◽  
pp. 201-202
Author(s):  
Keith Krehbiel
Keyword(s):  
Critical Values
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