A Special Note on the Logistic Functions with Complex Parameters and Some of Related Implications
HÜSEYİN IRMAK
Department of Mathematics, Faculty of Science,
Çankırı Karatekin University, Uluyazı Campus,
TR-18100, Çankırı,
TURKEY
Abstract: - By this special note, certain necessary information pertaining to the logistic function together
with some of its special forms (with real parameters) will be firstly introduced, and some results consisting
of several differential inequalities associated with various versions of the complex logistic functions
will be then determined. In addition, a number of special implications concerning those results will be also
indicated.
Key-Words: - The complex plane, analytic functions, power series, the logistic function, the sigmoid function,
the modified sigmoid function
Received: October 11, 2022. Revised: April 19, 2023. Accepted: May 11, 2023. Published: May 22, 2023.
1 Introduction and Preliminary
Information
As it is well known, in written literature, especially,
in mathematical science, there are many special
functions. These functions, which possess a wide
range of applications, appear as functions with both
real independent variables and complex
independent variables. One of those functions is
the logistic function, or, more specifically, the
sigmoid function. In this special research, some
necessary-basic information about those special
functions (with real independent parameters) will
be firstly introduced, and for some of the possible
results of our main research, which will consist of
various theoretical information, some of the
numerous complex forms of those indicated
functions will be then concentrated.
We can now begin to present (or evoke)
various necessary information dealing with this
special investigation.
For those, we begin by introducing the logistic
function with a real variable (or parameter) (or the
sigmoid function with a real variable (or
parameter)). We denote this special function (with
real variable) by the following-equivalent
notations:
L
≡
L
,ℓ
;
and also define by the form given by
L
,ℓ
;
ℓ
,1
where, of course, the associated parameters
,ℓ
(ℓ0and (0) belong to the set
consisting of the real numbers, which are usually
expressed by the midpoint, the maximum value,
and the logistic growth rate (or steepness) of the
related function L, respectively. Notably, in
terms of these specific technical terms, we should
emphasize here that values greater than zero of the
parameters ℓ and are significant values that can
be reasonable for various applications of the
logistic function with real parameters.
As certain theoretical-historical information, the
important function just above was introduced in a
series of three basic papers by Pierre Francois
Verhulst between the years 1838 and 1847, who
devised it as a classical model of population growth
by adjusting the exponential growth model, under
the guidance of Adolphe Quetelet. For each of
their details, respectively, one may refer to the
primary works given by, [33], [34], [35] in the
references.
Most particularly, in the literature, it also
appeared as a more specialized form of the logistic
function L, which is generally denoted by the
notation given by
:L
,
0;
and also called the standard logistic function (or the
sigmoid function), being of the form given by
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1
1
,2
where the main variable x will belong to any
interval, determined by the relevant field of related
study, in the well-known set
.
Specially, as a great number of interesting
implications of various special functions (with
real (or complex) parameters), those two special
functions (just above) has an important part in
numerous applications in a range of scientific fields
including biology, ecology, biomathematics,
demography, chemistry, sociology, political
science, geoscience, economics, mathematical
psychology, linguistics, probability, statistics,
mathematics, artificial neural networks and so on.
For some different applications of them, as an
example, one may concentrate on the relevant
results in the earlier papers given by, [3], [4], [6],
[19], [20], [24], [25], [27], [29], [31], [32].
For both necessary information and the scope of
this investigation, there is a need to evoke the well-
known Taylor-Maclaurin series expansions of the
following-elementary functions with real variable
: ≔ and ≔
given by
11
2!⋯1
!⋯3
and
1⋯1⋯, (4)
respectively.
By means of the well-known series expansions
presented in (3) and (4), the following equivalent-
extensive relations:
L
≡
L
,ℓ
;
ℓ
≡
ℓ
∑
5
≡
ℓ
≡
ℓ
∑
!
6
≡∑
∑
!
7
can be easily constituted, where
ℓ∈
∗
≔0,
∈
∗
,
∈
∗
8
As has been informed above, we have mentioned
some application areas of both the function L
and the function
, and we have also offered
numerous basic references for those.
However, since the basic purpose of this special
research will relate to certain complex forms of
those functions indicated above, some extra
information and also references about both certain
functions and special functions with complex
variables will also be needed. For more
information about both the relevant special
functions, introduced by (1) and (2), and,
especially, possible complex forms of those
functions, the main works given by the references
in, [2], [5], [7], [9], [13], [15], [16], [21], [22], [25],
[28], [38], [41] can be recommended to concerned
readers as main works (or scientific researchers).
Furthermore, under some extra conditions
associated with the mentioned parameters similar to
(8) and for several relations in relation to various
special functions (with complex parameters), a
number of extra researches, given by, [1], [10],
[12], [18], [22], [30], [37], [40], can be also
checked as different type investigations for
interested researchers.
2 Related Lemma and Main Results
In the previous section, we presented various basic
information and numerous special results about the
logistic function and its special forms with real
variables (or parameters). In this section, the
conditions in which some special cases of the
conditions specified in (8) will be determined, by
considering all possible-complex variable
structures in relation to those relevant special
functions, which are similar forms between (1)-(7),
several new (special) results regarding those
complex functions can be determined. Here, only
two of those new results will be also constituted.
As it is well known, by making use of the
important assertions presented by [14], the
following-useful lemma was proven in the paper
given by, [23], which will be needed for stating and
also proving each one of our main results. At the
same time, for extra information, it can be also
focused on the earlier works given by, [8], [17],
[26].
Lemma 1. Let ≔1,2,3,…and also let
1
⋯∀∈;∈9
be an analytic function in the open unit disk:
≔∶∈and||1,
and suppose that there exists a point ω0 belonging
to such that
0 for ||||10
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and 0.11
Then, the inequality:
1||12
is satisfied.
In light of the information indicated right after
the second part and the equivalent forms of the
functions consisting of real parameters between )5 (
and (7), if we consider the logistic function (with
complex variable ), which has the same notation
in the mentioned form:
L
≡
L
,ℓ
;
,13
various new results relating to both these new-
special functions consisting of complex mentioned
parameters and their special forms can be also
constituted, where
∈,
0
and
ℓ
∈
∗
.
14
Since each one of the complex functions like the
forms as in (13), which consists of any one of the
same forms between (5)-(7), is an important
function with the same properties as in the form in
Lemma 1, the mentioned lemma will be interesting
for stating and then proving each one of main
results of this paper.
Let us now present some of our main results and
then prove them by considering Lemma 1. As the
first main result, by considering the form with
complex parameters of the expression given in (6),
the following theorem, Theorem 1, can be then
constituted.
Theorem 1. Under the conditions specified by the
admissible values of the parameters emphasized in
(14), for the equivalent complex functions
presented by (13), if
ℓ
ℓ
ℓ
ℓ
15
holds, then
ℓ
1016
holds, where
∈.
Proof. For the related proof, we want to use
lemma 1. Therefore, first of all, of course, we need
to describe a complex function like that is as
in the relevant lemma and also has the series-
expansion form in (9). For it, under the mentioned
conditions in (14), let us consider the implicit form
consisting of both the function situated as in
Lemma and the complex logistic function indicated
in (13), which is
L
ℓ
∈
.17
Then, in consideration of the relation having the
same form presented by (1), for a suitable-analytic
function, the function can be taken as the
necessary form:
,
where
0
and∈ ,
or, equivalently, when the series expansion
possessing the same form given in (6) is considered
here, it also has the complex series expansion given
by
∑
!
,
where
0
and∈.
As a result of a simple focus, clear, both is an
analytic function in disc U and satisfies the
condition 01. Namely, it is one of the main
conditions of Lemma 1.
We next suppose that there exists a point ∈
, which satisfies the conditions given by (10) and
(11) of the lemma. By means of the implicit
function defined in (17), we then get that
ℓ
ℓ
≔
≔
||1
18
||1
||1
,
and, by using the mentioned conditions in (10) and
(11) of Lemma 1 for the result determined by (18),
we then arrive at
ℓ
ℓ
≔
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||1
||
||
||
or, equivalently, with help of the implicit function
formed by (17), we next derive that
ℓ
ℓ
≔
||
||
ℓ
ℓ
,
which also is inconsistent with the hypothesis of
Theorem 1 in equation (15). Therefore, in
accordance with
01
,0Θ2
and 0
2,
one can easily arrive at
cos
sinΘ
0,
which immediately requires the inequality given by
(16). So, this also ends the desired proof.
As the second main result, by considering the form
with complex parameters of the expression given in
(7), the following theorem, Theorem 2, can be also
composed.
Theorem 2. Let ∈ and ∈∶∪.
Under the conditions specified by the admissible
values of the parameters established by (14), for the
equivalent complex functions presented by (13), if
any one of the cases constituted by the conditions:
ℓ
1
ℓ∈
1
ℓ∈ (19
holds, then
ℓ020
holds, where the value of the complex power stated
in (19) (or in (20)) is considered as its principal
value, and, of course, the mentioned notations:
and denote the set of positive integers and
the set of negative integers, respectively.
Proof. First of all, let ∈ and ∈, and also
let the mentioned conditions satisfy those complex
parameters given by (14). Then, for its pending
proof, in view of the relation in (7) and by
considering an analytic function being of the
form given by
ℓ∑
∑
!
,21
Lemma 1 can be considered again as it was used in
the proof of Theorem 1. Because this defined
function with complex variable, namely,
satisfies the hypotheses given in Lemma 1. It also
follows from (21) that
ℓ
ℓ
∈.22
From here on, the similar ways, which have been
followed in the proof of Theorem 1, are then
considered for the equation obtained in (22) and if
Lemma 1 is also used there, all necessary steps of
the desired proof can be easily constructed.
Consequently, its proof is omitted here and its
detail is also brought to the attention of interested
researchers.
3 Conclusion and Recommendations
In this last section, we want to put emphasis on
certain special information. In this present research,
it is clearly seen that the fundamental theorem of
our research is directly related to the theory of
complex functions. For the emphasized
information, one can center on the references in,
[2], [8], [14], [25], [26], [36], [37].
As the first implication, especially, we have
focused on only two of any number of the possible
theories relating to the logistic function with
complex parameter by considering the basic
relations regarding the logistic function with real
parameter set out in the equivalent-equations
presented by (1), (5), (6) and (7). These presented
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theories are only two important results and many
new theories can be also created by making use of
the main-highlighted relations, which will be two
interesting examples for interested researchers.
As the next implications, all right, various
special (or remodified) results of the logistic
function can be also determined. For those, of
course, various type sigmoid functions with the
complex variable, which are some special cases of
the special function with real parameters introduced
by (2), can be also concentrated as certain
interesting implications. Those modified functions
with complex variables, are frequently encountered
in the mathematical literature and include various
different results in many different fields. For some
of them, especially, see also the papers in, [5], [6],
[10], [11], [12], [13], [18], [28], as certain special
examples. At the same time, for other possible
implications relating to those complex functions, it
is enough to select the values of the mentioned
parameters contained in the definition in (13), of
course, under the conditions accentuated in (14).
For extra information relating to various
possible results of those special functions with real
(or complex) variable, the research works presented
in, [2], [8], [12], [15], [20], [23], [25], [26], [34]
can also be as some main works for related
researchers. These extensive results are only two
important implications of our main results and they
also associate with Theorem 1. In addition, of
course, to reveal other extra new special
implications concerning our main results, it will be
enough to consider the appropriate parameters used
in their respective theorems. Nevertheless, as a
special implication of our investigation concerning
the sigmoid function with complex variables, we
want to present it as a proposition.
When taking account of the special relationship
between the logistic function (with real parameters)
in (1) and the sigmoid function (with real
parameters) in (2), of course, naturally, there is a
matter of the following important relationships
between the logistic function and certain
exponential type functions, which are the special
functions with complex variable given by the
following forms:
≡L
,
0;
(23)
and
≡L
,
0;
(24)
where ∈. We remark that the function
is known as the modified Sigmoid function
in the literature.
As additional extra information, it is clear that
when considering the complex exponential
function:
,
the following function:
cos
≡
cos
depends upon the sign of the trigonometric function
with real variable given by the form:
.
Naturally, in terms of the corresponding
complex exponential function , which will
play an important role both in the complex logistic
function and also in its special forms, the real
parameter and the imaginary parameter change
in the open interval (-1,1) when the complex
parameter z changes in the open set.
Indeed, since
,∈1,1⊂
,
⇒0andcos025
for all ∈,it is obvious that
cos0.26
In special, we should point out here that the
complex functions:
and
are also known as, respectively, the (complex)
sigmoid function and the modified (complex)
sigmoid function in the literature. For both
functions above, we think that the earlier results,
cited in, [12], [18], [19], [28], [30], [37], [39], [40],
in the references, are interesting papers containing
various analytical-geometric results of functions
with complex variables. So, for those special
functions presented in (23) and (24), by leaving the
details of the results to the interested readers, we
just want to present a new special implication
regarding the complex function given in (23).
For only one of the indicated implications of
Theorem 1, in consideration of the information in
(7) (or (6)) along with (25) and (26), by means of
Theorem 1 and also by choosing the values of the
parameters as ∶1, the value of the parameter
ℓ as ℓ := 1 and the value of the parameter as
:= 0 there, namely, by using the special function
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with complex variable composed as in (23), which
is the complex-sigmoid function being also of the
series expansion given by
≡
⋯ ,
the following-special proposition can be easily
created as only one of a great number of the
possible implications of Theorem 1.
Proposition 1. Let ∈. Then, the assertion:
|
|⇒
0
holds true.
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Contribution of Individual Authors to the
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Policy)
The author contributed to the present research, at
all stages from the formulation of the problem to
the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This research received no external funding.
Conflict of Interest
The authors have no conflict of interest to declare.
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