J. Math. Comput. Sci. 8 (2018), No. 2, 270-277 https://doi.org/10.28919/jmcs/3593
ISSN: 1927-5307
SINGULAR VALUES OF THREE-PARAMETER FAMILIES λ(e
az−1
z )µ AND λ(eazz−1)η
MOHAMMAD SAJID
College of Engineering, Qassim University, Buraidah, Al-Qassim, Saudi Arabia
Copyright c2018 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.
Abstract. The main goal of the present paper is to investigate the singular values of three-parameter families of
transcendental (i) entire functions fλ,a,µ(z) =λ
eaz−1 z
µ
and fλ,a,µ(0) =λaµ; µ>0, λ,a∈R\{0}, z∈C
(ii) meromorphic functionsgλ,a,η(z) =λ
z eaz−1
η
and gλ,a,η(0) = aλη,η >0; λ,a∈R\{0},z∈Cˆ. It is
obtained that all the critical values of fλ,a,µ(z)andgλ,a,η(z)lie in the right half plane fora<0 and in the left half
plane fora>0. It is also shown that all these critical values of fλ,a,µ(z)andgλ,a,η(z)are interior and exterior
of the open disk centered at origin and having radii|λaµ|and|λ
aη|respectively. Further, it is described that the functions fλ,a,µ(z)andgλ,a,η(z)both have infinitely many singular values.
Keywords:Critical values; singular values; entire function; meromorphic function.
2010 AMS Subject Classification:37F10, 58K05.
1. Introduction
The importance of singular values in the dynamics of transcendental functions can be seen
in [1, 4, 7, 17]. The dynamics of one-parameter familyλez, that has only one singular value, is
studied in detail, for instance see [2, 3]. This exponential family is simpler than other families
E-mail address: msajd@qu.edu.sa
Received November 27, 2017
which have more than one or infinitely many singular values. Often it is crucial to investigate the
dynamical properties of such functions. Some dynamical properties of families of such kinds of
functions associated with exponential map are found in [5, 6, 8, 9, 10, 15]. These investigations
are enormously applicable for describing Julia sets and Fatou sets.
This work investigates the singular values of three-parameter families of transcendental entire
and transcendental meromorphic functions. For this purpose, let us consider three-parameter
families of transcendental entire and meromorphic functions respectively which are neither even
nor odd and not periodic:
E =
fλ,a,µ(z) =λ
eaz−1
z
µ
and fλ,a,µ(0) =λaµ:µ>0,λ,a∈R\{0},z∈C
M =
gλ,a,η(z) =λ
z
eaz−1
η
andgλ,a,η(0) = λ
aη :η >0,λ,a∈R\{0},z∈
ˆ C
The familyE is a generalization of families of entire functionsλe
z−1
z [5],λ
eaz−1
z [14] and
λ
ez−1
z
m
[12]. The family M is a generalization of families of meromorphic functions
λ
z
ez−1[11],λ
z
eaz−1 [14] andλ
z
ez−1
m
[13]. In spite of these, assuminga=1, our families
of functions are associated to the well known generalized (i) Bernoulli’s generating function
z ez−1
α
etz=∑∞k=0B
(α)
k (t) zk
k! by choosing α >0 andt =0(ii) Apostol-Bernoulli’s generating
function z
λez−1
α
etz=∑∞k=0Bk(α)(t;λ)z
k
k! by choosingα>0,λ =1 andt=0 [16].
A pointz∗is said to be a critical point of f(z)if f0(z∗) =0. The value f(z∗)corresponding
to a critical point z∗ is called a critical value of f(z). A point w∈Cˆ = C∪ {∞} is said to
be an asymptotic value for f(z), if there exists a continuous curve γ : [0,∞)→Cˆ satisfying
limt→∞γ(t) =∞and limt→∞f(γ(t)) =w. A singular value of f is defined to be either a critical
value or an asymptotic value of f.
In Theorem 1, it is seen that the functions f0
λ,a,µ(z)andg
0
λ,a,η(z)have no zeros in the left half
plane and the right half plane fora<0 anda>0 respectively. It is found that, in Theorem 2,
the functions fλ,a,µ ∈E and gλ,a,η ∈M map (i) the right half plane interior and exterior of
the open disk fora<0 respectively (ii) the left half plane interior and exterior of the open disk
fora>0 respectively. In Theorem 3, it is shown that all the critical values of fλ,a,µ ∈E and
and|λ
aη|respectively. Further, it is described that the functions fλ,a,µ∈E andgλ,a,η ∈M have
infinitely many singular values in Theorem 4.
2. Singular Values of
f
λ,a,µ∈
E
and
g
λ,a,η∈
M
Let us denote the right half and left half planes byH+={z∈Cˆ :Re(z)>0}andH−={z∈
ˆ
C:Re(z)<0}respectively.
Lemma 1. The equation e−w=1−w has no solutions in the right half plane H+.
The rigorous proof of this lemma can be seen in [5] and a short proof is given in [11] by using
graphs.
Let Dr(0) be the open disk centered at origin and radiusr. In the following theorem, it is
found that the functions fλ,a,µ∈E andgλ,a,η ∈M have no critical points in the left half plane
and the right half plane fora<0 anda>0 respectively:
Theorem 1. Let fλ,a,µ∈E and gλ,a,η ∈M. Then,
(a) for a<0, f0
λ,a,µ(z)and g
0
λ,a,η(z)have no zeros in the left half plane H
−.
(b) for a>0, f0
λ,a,µ(z)and g
0
λ,a,η(z)have no zeros in the right half plane H
+.
Proof. We have f0
λ,a,µ(z) =λ µ( eaz−1
z )
µ−1(az−1)eaz+1
z2 forz6=0 and fλ0,a,µ(0) =λ µ
aµ+1
2 . Then,
for critical points of fλ,a,µ(z), we geteaz=1 ande−az=1−az. Hence, the zeros of f0
λ,a,µ(z)
arez= 2pπia on imaginary axis, where p is nonzero integer; and the solutions of the equation
e−az=1−az.
Similarly as above, we have g0
λ,a,η(z) =λ η( z eaz−1)η−1
(1−az)eaz−1
(eaz−1)2 forz6=0 andg0λ,a,η(0) =
−λ η 1
2aη−1, so that for zeros ofg0λ,a,η(z), we gete−az=1−az.
Letw=az. Then,e−w=1−wfor above both cases. By Lemma 1, this equation has no any
solution inH+.
Therefore, it shows that both the functions f0
λ,a,µ(z) andg
0
λ,a,η(z) have no zeros inH
− for
Lemma 2. Let h(z) =eaz for an arbitrary fixed z∈C. Then, for (i) a<0 and z∈H+, and
(ii) a>0and z∈H−, the following inequality holds
|eaz−1|<|z||a| (1)
Proof.Suppose that the line segmentγ is defined byγ(t) =tz,t∈[0,1].
SinceM≡maxt∈[0,1]|h(γ(t))|=maxt∈[0,1]|(ea)tz|<1 fora<0 andz∈H+, then
Z
γ
h(z)dz=
Z 1 0
h(γ(t))γ0(t)dt=z Z 1
0
eatzdt =1
a(e
az−1)
|eaz−1|=a
Z
γ
h(z)dz≤M|z||a|<|z||a|
Using similar arguments as above, we can easily deduce the inequality fora>0.
It is shown in the following theorem that the functions fλ,a,µ ∈E andgλ,a,η ∈M map the
right half plane and the left half plane interior and exterior of the open disk respectively:
Theorem 2. Let fλ,a,µ∈E and gλ,a,η ∈M. Then,
(i) for a <0, fλ,a,µ(z) and gλ,a,η(z) map the right half plane H+ interior D|λaµ|(0) and
exterior of D|λ
aη|(0)
respectively.
(ii) for a>0, fλ,a,µ(z)and gλ,a,η(z)map the left half plane H−interior D|λ|aµ(0)and exterior
of D|λ| aη
(0)respectively.
Proof.(i)Fora<0, using Inequality (1) of Lemma 2, we have
eaz−1
z
<|a| for allz∈H+
eaz−1
z µ =
eaz−1
z µ
<|a|µ since µ >0
|fλ,a,µ(z)|= λ
eaz−1
z
µ
<|λaµ| for allz∈H+.
Again, from Inequality (1), we get z
eaz−1
> 1
|a| for allz∈H
+ z
eaz−1
η = z
eaz−1
η > 1
|a|η sinceη >0
|gλ,a,η(z)|= λ z
eaz−1
η > λ aη
for allz∈H+.
This shows that the functiongλ,a,η ∈M mapsH+exterior ofD|λ
aη|
(0).
(ii)The proof of this part can be obtained similar as part(i)by using Inequality (1) of
Lem-ma 2(ii), hence it is omitted. This completes the proof of theorem.
In the following theorem, it is proved that the functions fλ,a,µ ∈E and gλ,a,η ∈M have
critical values interior and exterior of the open disk respectively:
Theorem 3. Let fλ,a,µ ∈E and gλ,a,η ∈ M. Then, all the critical values of fλ,a,µ(z) and
gλ,a,η(z)lie interior of D|λaµ|(0)and exterior of D|λ
aη|
(0)respectively.
Proof.Fora<0, by Theorem 1(a), the function f0
λ,a,µ(z)has no zeros inH
−. Therefore, all the
critical points of fλ,a,µ(z)lie inH+. By Theorem 2(i), the function fλ,a,µ(z)mapsH+ interior
ofD|λaµ|(0). It gives that all the critical values of fλ,a,µ∈E are lying interior ofD|λaµ|(0).
Fora>0, by Theorem 1(b), the function f0
λ,a,µ(z)has no zeros inH
+. Hence, all the critical
points of fλ,a,µ(z) lie in H−. By Theorem 2(ii), the function fλ,a,µ(z) maps H− interior of
D|λ|aµ(0). It follows that all the critical values of fλ,a,µ ∈E are lying interior ofD|λ|aµ(0).
Similarly, using analogous arguments as above, we can obtain the proof that all the critical
values of gλ,a,η ∈M lie exterior of D|λ
aη|
(0) for both a<0 and a>0. This completes the
proof.
Lemma 3. The equation sinv(v)−evcot(v)−1=0has infinitely many solutions.
The proof of this lemma can be seen in [5, 11].
The following theorem shows that the functions fλ,a,µ∈E andgλ,a,η ∈M possess infinitely
Theorem 4. Let fλ,a,µ ∈E and gλ,a,η ∈M. Then, the functions fλ,a,µ(z)and gλ,a,η(z)have infinitely many singular values.
Proof. For critical points of fλ,a,µ ∈E, we have f0
λ,a,µ(z) =0. Then, we gete
−az=1−azand
z=2pπia , wherepis nonzero integer. Note that fλ,a,µ(2pπa i) =0. Similarly, for critical points of
gλ,a,η ∈M,g0
λ,a,η(z) =0, we gete
−az=1−az.
Now, separating the real and imaginary parts ofe−az=1−az, then
ay
sin(ay)−e
aycot(ay)−1=0 (2)
x= 1
a−ycot(ay) (3)
Substitutingv=ayin Equation (2),
v
sinv−e
vcotv−1=0 (4)
By Lemma 3, Equation (4) has infinitely many roots. Therefore, it follows that Equation (2) has
infinitely many solutions for botha<0 anda>0. Let{yk}k=∞
k=−∞,k6=0be the infinitely solutions
of Equation (2). Then, from Equation (3),xk=1a−ykcot(ayk)fork=±1,±2,±3, . . ..
For critical points zk =xk+iyk of fλ,a,µ(z) and gλ,a,η(z), the critical values fλ,a,µ(zk) =
λ
eazk−1
zk
µ
andgλ,a,η(zk) =λ
zk
eazk−1
η
are distinct for distinctk. It shows that fλ,a,µ∈
E andgλ,a,η ∈M have infinitely many critical values fora<0 anda>0.
The point 0 is the finite asymptotic value of fλ,a,µ ∈E andgλ,a,η ∈M since fλ,a,µ(z)→0
as z→∞ along both positive and negative real axes for a< 0 and a> 0 respectively; and
gλ,a,µ(z)→0 asz→∞along both negative and positive real axes fora<0 anda>0
respec-tively.
Thus, the functions fλ,a,µ ∈E and gλ,a,η ∈M have infinitely many singular values. This
completes the proof.
Conclusion
In the present paper, the singular values of three-parameter families of transcendental entire
values of both families lie interior and exterior of the disk centered at origin and having finite
radii respectively. We have also shown that both the families of transcendental functions possess
infinitely many singular values.
Conflict of Interests
The author declares that there is no conflict of interests.
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