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On Certain Properties of a Degenerate Sigmoid Function

Thomas Awinba Akugre1,∗ , Kwara Nantomah2 , Mohammed Muniru Iddrisu3
1Department of Mathematics, School of Mathematical Sciences, C. K. Tedam University of Technology and

Applied Sciences, P. O. Box 24, Navrongo, Upper-East Region, Ghana
takugre.stu@cktutas.edu.gh

2Department of Mathematics, School of Mathematical Sciences, C. K. Tedam University of Technology and
Applied Sciences, P. O. Box 24, Navrongo, Upper-East Region, Ghana

knantomah@cktutas.edu.gh
3Department of Mathematics, School of Mathematical Sciences, C. K. Tedam University of Technology and

Applied Sciences, P. O. Box 24, Navrongo, Upper-East Region, Ghana
middrisu@cktutas.edu.gh

∗Correspondence: takugre.stu@cktutas.edu.gh

Abstract. In this paper, we introduce a degenerate sigmoid function. By employing analytical tech-niques, we present some properties such as logarithmic concavity, monotonicity and inequalities ofthe new function.

1. Introduction
It is known that, what is currently referred to as the logistic equation or the S-shaped curve wasfirst introduced by Verhulst (see [17]). It maps a very large input domain to a small range of outputof real numbers between 0 and 1. It is a one - to- one functioon and increases monotonically(see [8]). The sigmoid function, also known in the literature as the Sigmoidal curve or standardlogistic function is defined as (see [13]),

S (t) =
et

1 + et
=

1

1 + e−t
, t ∈ (−∞,∞) , (1)

=
1

2
+

1

2
tanh

(
t

2

)
, t ∈ (−∞,∞) . (2)

It has the following as its first and second derivatives
Received: 27 Feb 2023.
Key words and phrases. degenerate sigmoid function; logarithmically concave; inequality.1

https://adac.ee
https://doi.org/10.28924/ada/ma.3.17
https://orcid.org/0009-0005-1387-377X
https://orcid.org/0000-0003-0911-9537
https://orcid.org/0000-0001-7628-8168


Eur. J. Math. Anal. 10.28924/ada/ma.3.17 2

S
′
(t) =

et

(1 + et)2
= S (t) (1− S (t)) , (3)

S
′′

(t) =
et
(

1− et
)

(1 + et)3
= S (t) (1− S (t)) (1− 2S (t)) , (4)

for all t ∈ (−∞,∞) .The sigmoid function is used in a wide range of scientific disciplines, including probability andstatistics, biology, demography, machine learning, population dynamics, ecology, and mathematicalpsychology(see [7], [16]). In the business sector, the sigmoid function has been utilized to analyzeperformance growth in manufacturing and service management (see [10]). At each neuron’s output,the function serves as an activation function in artificial neural networks (see [12], [18], [15]) andthe references therein.In addition, the function is used in medicine to research pharmacokinetic responses and mimictumor development (see [11]). In [5], the site index of unmanaged loblolly and slash pine plantationsin East Texas is predicted using a generic variant of the sigmoid function. It is also used incomputer graphics and image processing to improve picture contrast (see [4], [9], [6]). It is clearfrom the above applications of the sigmoid function that, further research needs to be conductedon this very important function to unearth more of its properties and potential applications.
Recently, in [13], the author studied properties such as super multiplicativity, subadditivity,convexity and inequalities of the sigmoid function.
In this paper, a degenerate sigmoid function is introduced and properties such as logarithmicconcavity, monotonicity and inequlities involving the function are provided. We start with thefollowing definitions and lemmas.

2. Some Definitions and Lemmas
Definition 2.1. [1] A function M : (0,∞)×(0,∞)→ (0,∞) is called a mean function if it satisfiesthe following.(1) M (r, t) = M (t, r) ,(2) M (t, t) = t,(3) r < M (r, t) < t, for r < t,(4) M (ηr, ηt) = ηM (r, t) , for η > 0.

There are many well-known mean functions in the literature. Amongst them are the following.(1) Arithmetic mean: A (r, t) = r+t
2 ,(2) Geometric mean: G (r, t) =
√
r t,

https://doi.org/10.28924/ada/ma.3.17


Eur. J. Math. Anal. 10.28924/ada/ma.3.17 3(3) Harmonic mean: H (r, t) = 1
A( 1r ,

1
t )

= 2r t
r+t ,(4) Logarithmic mean: L (r, t) = r−t

ln r−ln t , for r 6= t and L (t, t) = t,(5) Identric mean: I (r, t) = 1
e

(
r r

tt

) 1
r−t , for r 6= t and I (t, t) = t.

Definition 2.2. [1] Let g : I ⊆ (0,∞)→ (0,∞) be a continuous function and U and V be any twomean functions. Then, g is said to be UV −convex (UV −concave) if
g (U (r, t)) ≤ (≥) V (g (r) , g (t)) ,

for all r, t ∈ I.
Lemma 2.3. [1] Let f : I ⊆ (0,∞)→ (0,∞) be a differentiable function. Then(1) f is AG-convex(or concave) if and only if f

′
(t)
f (t) is increasing(or decreasing) for all t ∈ I .(2) f is AH-convex( or concave) if and only if f
′
(t)

f (t)2
is increasing(or decreasing) for all t ∈ I .

Lemma 2.4. [2] Let f : I ⊆ (b,∞)→ (−∞,∞) with b ≥ 0. If the function defined by g (t) = f (t)−1
t

is increasing on (b,∞) , then the function h (t) = f
(
t2
)

satisfies the Grumbaum-type inequality

1 + h
(
z2
)
≥ h

(
r2
)

+ h
(
t2
)
, (5)

where r, t ≥ b and z2 = r2 + t2. If g is decreasing, then the inequality (5) is reversed.

3. Main Results
Definition 3.1. The degenerate sigmoid function is defined for λ ∈ (0,∞) and t ∈ (−∞,∞) as

Sλ (t) =
(1 + λt)

1
λ

1 + (1 + λt)
1
λ

(6)
=

1

1 + (1 + λt)−
1
λ

(7)
=

1

2
+

1

2
tanhλ

(
t

2

)
. (8)

It is clear that, taking the limit of Sλ (t) as λ→ 0, then Sλ (t)→ S (t) .The first derivative of the degenerate sigmoid function is given as
S
′
λ (t) =

(1 + λt)
1
λ
−1[

1 + (1 + λt)
1
λ

]2 > 0, (9)
for all t ∈ (−∞,∞) and λ ∈ (0,∞) .

The degenerate sigmoid function satisfies the following identities.

https://doi.org/10.28924/ada/ma.3.17


Eur. J. Math. Anal. 10.28924/ada/ma.3.17 4

Sλ (t) + Sλ (−t) = 1, (10)
Sλ (t)Sλ (−t) = (1 + λt)S

′
λ (t) , (11)

S
′
λ (t) = S

′
λ (−t) , (12)

lim
t→∞

Sλ (t) = 1, (13)
lim
t→0

Sλ (t) =
1

2
, (14)

lim
t→0

S
′
λ (t) =

1

4
, (15)

lim
t→∞

S
′
λ (t) = 0. (16)

Theorem 3.2. The function Sλ (t) is AG-concave on (0,∞). In other words, for all r, t, λ ∈ (0,∞) ,

the inequality

Sλ

(
r + t

2

)
≥ [Sλ (r)Sλ (t)]

1
2 (17)

is satisfied.

Proof. We have
S
′
λ (t)

Sλ (t)
=

 (1 + λt)
1
λ
−1[

1 + (1 + λt)
1
λ

]2
(1 + (1 + λt)

1
λ

(1 + λt)
1
λ

)

=
1

(1 + λt) + (1 + λt)
1
λ
+1

and (
S
′
λ (t)

Sλ (t)

)′
= −

λ+ (1 + λ) (1 + λt)
1
λ[

(1 + λt) + (1 + λt)
1
λ
+1
]2 < 0, (18)

which imlplies that S
′
λ(t)

Sλ(t)
is decreasing on (0,∞). Hence, by Lemma 2.3(1), we obtain the desiredresult (17). �

Theorem 3.3. The function Sλ (t) is AH-concave on (0,∞). In other words, for all r, t, λ ∈ (0,∞) ,

the inequality

Sλ

(
r + t

2

)
≥

2Sλ (r)Sλ (t)

Sλ (r) + Sλ (t)
(19)

is valid.

https://doi.org/10.28924/ada/ma.3.17


Eur. J. Math. Anal. 10.28924/ada/ma.3.17 5

Proof. Now we have
S
′
λ (t)

Sλ (t)2
=

 (1 + λt)
1
λ
−1[

1 + (1 + λt)
1
λ

]2


[

1 + (1 + λt)
1
λ

]2
(1 + λt)

2
λ


=

1

(1 + λt) (1 + λt)
1
λ

=
1

(1 + λt)
1
λ
+1

and (
S
′
λ (t)

Sλ (t)2

)′
= −

(1 + λ) (1 + λt)
1
λ

(1 + λt)
2
λ
+2

< 0.

By Lemma 2.3(2), we conclude that Sλ (t) is AH-concave on (0,∞). This implies inequality(19). �

Theorem 3.4. The function Sλ (t), for r, t, λ ∈ (0,∞) and z2 = r2+t2, satisfies the Grunbaum-type
inequality

1 + Sλ
(
z2
)
≥ Sλ

(
r2
)

+ Sλ
(
t2
)
. (20)

Proof. Let h (t) be defined for t, λ ∈ (0,∞) as h (t) = Sλ(t)−1
t . This implies

h (t) =

(1+λt)
1
λ

1+(1+λt)
1
λ
− 1

t

=−
1

t + t (1 + λt)
1
λ

.

Differentiating h (t), we have
h
′
(t) =

1 + (1 + λt)
1
λ + t (1 + λt)

1
λ
−1[

t + t (1 + λt)
1
λ

]2 > 0,

which implies that h (t) is increasing. By applying Lemma 2.4, we obtain the desired result (20). �
Theorem 3.5. For λ ∈ (0,∞) , the function Sλ (t) satisfies the inequalities

S2λ (r + t) ≥ Sλ (r)Sλ (t) , r, t ∈ [0,∞) (21)
and

S2λ (r + t) ≤ Sλ (r)Sλ (t) , r, t ∈ (−∞, 0] . (22)
Equality holds if r = t = 0.

https://doi.org/10.28924/ada/ma.3.17


Eur. J. Math. Anal. 10.28924/ada/ma.3.17 6

Proof. Let r, t ∈ [0,∞) and λ ∈ (0,∞). Recall that Sλ (t) is increasing. Thus we have
Sλ (r + t) ≥ Sλ (r) > 0, (23)
Sλ (r + t) ≥ Sλ (t) > 0, (24)

since r + t ≥ r and r + t ≥ t. Now by multiplying (23)and (24), we obtain the desired result (21).Next, let r, t ∈ (−∞, 0] and λ ∈ (0,∞), we have
0 < Sλ (r + t) ≤ Sλ (r) , (25)
0 < Sλ (r + t) ≤ Sλ (t) , (26)

since r + t ≤ r and r + t ≤ t. By multiplying the inequalities (25) and (26), we have the desiredresult. �

Theorem 3.6. The function Sλ (t) , for λ ∈ (0,∞) , satisfies the inequalities

S2λ (r t) ≤ Sλ (r)Sλ (t) , r, t ∈ [0, 1] (27)
and

S2λ (r t) ≥ Sλ (r)Sλ (t) , r, t ∈ [1,∞) . (28)
Equality holds if r = t = 1.

Proof. Let r, t ∈ [0, 1] and λ ∈ (0,∞). Recall that Sλ (t) is increasing. Thus we have
0 < Sλ (r t) ≤ Sλ (r) , (29)
0 < Sλ (r t) ≤ Sλ (t) , (30)

since r t ≤ r and r t ≤ t. Now by multiplying (29)and (30), we obtain the result (27).Next, let r, t ∈ [1,∞, ) and λ ∈ (0,∞), we have
Sλ (r t) ≥ Sλ (r) > 0, (31)
Sλ (r t) ≥ Sλ (t) > 0, (32)

since r t ≥ r and r t ≥ t. By multiplying the inequalities (31) and (32), the desired result isobtained (28). �

Theorem 3.7. For r, t ∈ (−∞,∞) and λ ∈ (0,∞), the function Sλ (t) is logarithmically concave.
In other words, the inequality

Sλ

(
r

a
+
t

b

)
≥ [Sλ (r)]

1
a [Sλ (t)]

1
b (33)

is satisfied. Where a > 1 and 1a + 1
b = 1.

https://doi.org/10.28924/ada/ma.3.17


Eur. J. Math. Anal. 10.28924/ada/ma.3.17 7

Proof. Let Q (t) = lnSλ (t) . Then,

Q
′
(t) =

S
′
λ (t)

Sλ (t)
=

(1+λt)
1
λ
−1[

1+(1+λt)
1
λ

]2
(1+λt)

1
λ

1+(1+λt)
1
λ

=

 (1 + λt)
1
λ

(1 + λt)
[

1 + (1 + λt)
1
λ

]2
(1 + (1 + λt)

1
λ

(1 + λt)
1
λ

)

=
1

(1 + λt) + (1 + λt)
1
λ
+1
.

Taking the second derivative of Q (t) , we have
Q
′′

(t) =−
λ+ (1 + λ) (1 + λt)

1
λ[

(1 + λt) + (1 + λt)
1
λ
+1
]2 < 0,

and this completes the proof. �

Corollary 3.8. For λ ∈ (0,∞) and t ∈ (−∞,∞) , the inequalities

S
′′
λ (t)Sλ (t) ≤

[
S
′
λ (t)

]2 (34)
and

Sλ (1 + u)Sλ (1− u) ≤

[
(1 + λ)

1
λ

1 + (1 + λ)
1
λ

]2 (35)
are valid.

Proof. Since Sλ (t) is logarithmically concave, then [ln (Sλ (t))]
′′
≤ 0, for all t ∈ (−∞,∞) and

λ ∈ (0,∞) . This implies that,
[ln (Sλ (t))]

′′
=

[
S
′
λ (t)

Sλ (t)

]′
=
S
′′
λ (t)Sλ (t)− S′λ (t)S

′
λ (t)

[Sλ (t)]2

=
S
′′
λ (t)Sλ (t)−

[
S
′
λ (t)

]2
[Sλ (t)]2

≤ 0.

Hence, S′′λ (t)Sλ (t)−
[
S
′
λ (t)

]2 ≤ 0, which yields equation (34).

https://doi.org/10.28924/ada/ma.3.17


Eur. J. Math. Anal. 10.28924/ada/ma.3.17 8Next, let a = b = 2, t = 1 + u and r = 1− u in equation (33). we have
Sλ

(
1 + u

2
+

1− u
2

)
≥ [Sλ (1 + u)]

1
2 [Sλ (1− u)]

1
2

Sλ (1) ≥ ([Sλ (1 + u)] [Sλ (1− u)])
1
2[

(1 + λ)
1
λ

1 + (1 + λ)
1
λ

]2
≥ Sλ (1 + u)Sλ (1− u) ,

resulting in equation (35). This concludes the proof. �

Theorem 3.9. For t, λ ∈ (0,∞) , the function Sλ (t) satisfies the inequality

1 <
Sλ (t + 1)

Sλ (t)
<

2 (1 + λ)
1
λ

1 + (1 + λ)
1
λ

. (36)
Proof. Recall from equation (18), that(

S
′
λ (t)

Sλ (t)

)′
= −

λ+ (1 + λ) (1 + λt)
1
λ[

(1 + λt) + (1 + λt)
1
λ
+1
]2 < 0,

for all t, λ ∈ (0,∞) . This implies, the function S
′
λ(t)

Sλ(t)
is decreasing on the given interval.Now, let

P (t) =
Sλ (t + 1)

Sλ (t)
=

(
[1 + λ (t + 1)]

1
λ

1 + [1 + λ (t + 1)]
1
λ

)(
1 + (1 + λt)

1
λ

(1 + λt)
1
λ

)

=
[1 + λ (t + 1)]

1
λ + (1 + λt)

1
λ [1 + λ (t + 1)]

1
λ

(1 + λt)
1
λ + (1 + λt)

1
λ [1 + λ (t + 1)]

1
λ

and
Ω (t) = lnP (t)

= lnSλ (t + 1)− lnSλ (t) .

Then,
Ω
′
(t) =

S
′
λ (t + 1)

Sλ (t + 1)
−
S
′
λ (t)

Sλ (t)
< 0,

since S
′
λ(t)

Sλ(t)
is decreasing. This implies Ω (t) and consequently P (t) are decreasing. Hence, forall t, λ ∈ (0,∞) , we have

1 = lim
t→∞

P (t) < P (t) < lim
t→0

P (t) =
2 (1 + λ)

1
λ

1 + (1 + λ)
1
λ

,

which yields the desired result (36). �

https://doi.org/10.28924/ada/ma.3.17


Eur. J. Math. Anal. 10.28924/ada/ma.3.17 94. conclusion
We have introduced a degenerate sigmoid function. Properties such as concavity, monotonicity andinequalities involving the new function have been established. These established properties canbe applied in several areas of mathematics.

5. Conflicts of interest
The corresponding author affirms on behalf of all authors that there is no conflict of interest for thepublication of this research.

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	1. Introduction
	2. Some Definitions and Lemmas
	3. Main Results
	4. conclusion
	5. Conflicts of interest
	References

