EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 2349-2360
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Some Results on Mathai-Haubold Fuzzy Entropy

Vaishali Manish Joshi1,2, Javid Gani Dar1,∗

1 Department of Applied Sciences, Symbiosis Institute of Technology,
Symbiosis International(Deemed University), Pune 412115, India
2 Dr.Vishwanath Karad MIT World Peace University,Pune 411038, India

Abstract. The concept of entropy, emerged from thermodynamics and statistical mechanics is of
fundamental importance in some scientific and technological areas such as communication theory,
physics, probability and statistics. Fuzzy entropy is much looked upon concept for measuring fuzzy
information. The concept of fuzzy entropy was firstly mentioned by Zadeh way back in 1965 as
a measure of fuzziness. In this paper , we introduce Mathai - Haubold fuzzy entropy with the
proof of its validity. In addition, the elegant properties are studied of the proposed fuzzy entropy
measure.

2020 Mathematics Subject Classifications: 62B10 ,62R07,60G35, 62N05

Key Words and Phrases: Entropy, Generalized Entropy, Fuzzy Sets, Fuzzy Entropy, Shannon
Entropy, Renyi Entropy, Generalized Fuzzy Entropy, Mathai-Haubold Entropy

1. Introduction

Theory of information grew from the invention of telegraphs and telephones. To dealt
with the problems related to transmission of signal, many researchers contributed in this
field. Initially, Harry Nyquist [12],[13] given a formula to calculate the rate of finite band-
width in noiseless channel. Later on, Hartley [5] established the measure of information.
This measure is then modified by Claude Shannon [15], which is known as Shannon’s
Entropy. Shannon observed that the amount of information sent by a signal is inversely
proportional to the size of the message. In probability distribution, entropy takes max-
imum value when all probabilities are equal. The word entropy is defined as a measure
of uncertainty in a probability distribution. The concept of entropy is widely used in
many engineering applications such as clustering, image processing, statistical mechanics,
finance, neural networks, pattern recognition, in medical images for cells counting, fuzzy
clustering, speech recognition etc.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5304

Email addresses: vaishali.joshi.phd2023@sitpune.edu.in (Vaishali Manish Joshi),
javid.dar@sitpune.edu.in (J. G. Dar), vaishali.joshi@mitwpu.edu.in (V. M. Joshi)

https://www.ejpam.com 2349 © 2024 EJPAM All rights reserved.



V. M. Joshi, J. G. Dar / Eur. J. Pure Appl. Math, 17 (3) (2024), 2349-2360 2350

2. Properties of Entropy Function

In this section, we have covered some basic concepts like entropy, fuzzy sets and fuzzy
entropy which are required for the proposed work.

2.1. Basic Concepts of Entropy

Shannon’s entropy for a discrete random variable Q = {q1, q2, ..., qn} is defined as

D(Q) = −
n∑

k=1

qklogqk (1)

where qk is the probability associated with the event ek, for k = 1, 2, ..., n
Properties of Entropy Function:
a) Continuity: Entropy function D(Q) should be continuous.That means for every inde-
pendent variable 0 ≤ qk ≤ 1 , entropy function must be continuous.
b) Symmetry: Entropy function D(Q) remains unchanged when q1, q2, ..., qn are inter-
changed with each other.
c) Maximality: Entropy function D(Q) is maximum when all probabilities are equal.
d) Additivity Property: This property of D(Q) states that if a particular event xn with
probability qn is divided into m mutually exclusive subsets say e1, e2, ..., em with proba-
bilities r1, r2, ..., rm such that qn = r1 + r2 + ...+ rm then
D(q1, q2, ...qn, r1, r2, ..., rn) = D(q1, q2, ..., qn−1) + qnD( r1q1 ,

r2
q2
, ..., rnqn )

After the Shannon’s entropy measure, some of the listed generalizations were seen.
a) Renyi entropy [14] of order α

Dα(Q) =
1

1− α
log

n∑
k=1

qαk , α ̸= 1, α > 0 (2)

b) Havrda-Charvat [6] entropy of order α

Dα(Q) =
1

21−α − 1

n∑
k=1

qαk − 1, α ̸= 1, α > 0 (3)

c) Tsallis entropy [16] of order α

Dα(Q) =
1

1− α

n∑
k=1

qαk − 1, α ̸= 1, α > 0 (4)

d) Mathai-Haubold entropy [11] of order α

Dα(Q) =
1

α− 1

n∑
k=1

q2−α
k − 1, α ̸= 1,−∞ < α < 2 (5)



V. M. Joshi, J. G. Dar / Eur. J. Pure Appl. Math, 17 (3) (2024), 2349-2360 2351

Dα(Q) =
1

α− 1
log

n∑
k=1

q2−α
k , α ̸= 1,−∞ < α < 2 (6)

As α → 1 , above all the equations from (2) to (6) reduces to Shannon’s entropy.
Hence these are known as generalized entropies of order α.

2.2. Fuzzy Set

The concept of Fuzzy set theory of probability theory was proposed by Lofti Zadeh
[18], which achieved a big success in various fields. Zadeh introduced the concept of
fuzzy entropy as a measure of uncertainty due to the fuzziness in information. Kapur [8]
argued that the fuzzy entropy measures uncertainty due to fuzziness of information, while
probabilistic entropy measures uncertainty due to the information available in terms of
probability distribution only.
Fuzzy set is an extension of classical set, which is defined as B = {(x, ηB(x)/x ∈ X} with
the membership function of B as ηB : X → [0, 1]. The membership value gives the degree
of belongingness of an element x ∈ B. Here the end values 0 and 1 gives no membership
and full membership respectively. The membership function ηB(x) is defined as follows:

ηB(x) =


0 if x /∈ B and no ambiguity,

1 if x ∈ B and no ambiguity,

0.5 if max ambiguity, x ∈ B or not.

(7)

Fuzzy set operations are the generalizations of crisp set operations. Some operations on
fuzzy sets, which are required for our discussion, are as follows:
a) Union of fuzzy sets: Let R,S, T be fuzzy sets of universe of discourse Y , then union
operation is defined as:

R ∪ S = max(ηR(x), ηS(x)) (8)

((R ∪ S) ∪ T ) = {y ∈ Y, (y,max(max(ηR(y), ηS(y)), ηT (y))} (9)

b) Intersection of fuzzy sets: Let R,S, T be fuzzy sets of universe of discourse Y , then
intersection operation is defined as:

R ∩ S = min(ηR(x), ηS(x)) (10)

((R ∩ S) ∩ T ) = {y ∈ Y, (y,min(min(ηR(y), ηS(y)), ηT (y))} (11)

c) Complement of Fuzzy set: Let R be a fuzzy set, complement of R is defined as
ηRc(x) = 1− ηR(x).

2.3. Fuzzy Entropy

Entropy of a fuzzy set, as the probability measure of fuzzy information is defined by
Zadeh [17] and it is given as follows:

D(B) = −
n∑

k=1

ηB(xk)pklogd(pk) (12)



V. M. Joshi, J. G. Dar / Eur. J. Pure Appl. Math, 17 (3) (2024), 2349-2360 2352

where ηB represent the membership function of B and X= {x1, x2, ...., xn} be a discrete
random variable with probability distribution {p1, p2, ...., pn}. In (12), when the base
value d = 2, then the entropy measure is called as bit, for d = 10, it is known as Heartley
and for d = e, it is called as nat. Usually in the communication system the sourcecode
is converted to bit and hence we use logarithm to the base 2. In fuzzy set theory each
element is associated with the degree of membership, these membership values are lies
between 0 and 1 but they are not probabilities as their sum is not equal to 1. Hence
Kauffman [9] defined a fuzzy entropy of set B as

D(B) = − 1

logn

n∑
k=1

ψB(xk)log(ψB(xk)) (13)

where ψB(xk) = ηB(xk)∑n
k=1 ηB(xk)

is a probability distribution. It means that Fuzzy entropy

is nonprobabilistic entropy. A measure of fuzziness H(B) in a fuzzy set should have the
following four properties:
a) D(B) = 0 if and only if B is a crisp set.
For ηB(xi) = 0 or ηB(xi) = 1 , the value of D(B) is zero.
b) D(B) is maximum if B is most fuzzy set.
If ηB(xi) = 0.5 then D(B) takes the maximum value.
c) D(B∗) > D(B) where B∗ is a sharpened version of B.
d) D(Bc) =D(B) where Bc is the complement of fuzzy set B
As ηB(xi) and 1− ηB(xi) have same membership value, taking this into account De Luca
and Termini [10] introduced a new measure of Fuzzy entropy corresponding to Shannon’s
entropy

D(B) = −
n∑

k=1

ηB(xk)log(ηB(xk)) + (1− ηB(xk))log(1− ηB(xk)) (14)

Equation (14) satisfies all the four properties (a) to (d) , hence it is a valid measure of
fuzzy entropy. Later on Bhandari and Pal [2] and J. Kapur [8] suggested the following
measure of fuzzy entropy

D(B) = −
n∑

i=1

[ηB(xi)log(ηB(xi))
α + (1− ηB(xi))log(1− ηB(xi))

α] (15)

and

D(B) = −
n∑

i=1

(ηB(xi)
α + (1− ηB(xi))

α − 1) (16)

respectively.
In recent years, many researchers [3],[4],[7],[11],[1],[8] etc. have studied and introduced

several generalizations of fuzzy entropy measures. The remaining paper is organized as
follows. In Section 2, given the basic concepts of entropy function by covering the basic
terms like entropy, fuzzy sets and fuzzy entropy, required for the proposed generalization
of fuzzy entropy. In Section 3, we have proposed a new parametric generalized fuzzy



V. M. Joshi, J. G. Dar / Eur. J. Pure Appl. Math, 17 (3) (2024), 2349-2360 2353

entropy measure corresponding to [11]. Section 4 provides some more elegant properties
of the proposed measure in a number of theorems. Finally some concluding remarks in
Section 5.

3. Generalized Fuzzy Entropy of order α

Here we propose a new generalized fuzzy entropy measure of Mathai -Haubold entropy
corresponding to [11] and checked it’s validity.

Definition 1. Let B be the fuzzy set defined on X = {x1, x2, ....., xn} with the membership
values ηB(xi) for i = 1, 2, ..., n then the generalized fuzzy entropy of order α is defined as

Mα(B) =
1

n(2α−1 − 1)

n∑
i=1

[ηB(xi)
2−α + (1− ηB(xi))

2−α − 1], α ̸= 1, 0 < α < 2 (17)

Theorem 1. Mα(B) is a valid measure of fuzzy entropy.

Proof. To show Mα(B) a valid fuzzy entropy measure.
a) To check Mα(B) = 0 if and only if B is a crisp set. that is for ηB(xi) = 0 or ηB(xi) = 1
, the value of Mα(B) is zero. If ηB(xi) = 0 then the equation (16)

Mα(B) =
1

n(2α−1 − 1)

n∑
i=1

[ηB(xi)
2−α + (1− ηB(xi))

2−α − 1] (18)

is equal to zero.
If ηB(xi) = 1 then ,

Mα(B) =
1

n(2α−1 − 1)

n∑
i=1

[ηB(xi)
2−α + (1− ηB(xi))

2−α − 1] = 0 (19)

which is a minimum. Conversely if ,

Mα(B) =
1

n(2α−1 − 1)

n∑
i=1

[ηB(xi)
2−α + (1− ηB(xi))

2−α − 1] = 0 (20)

then easily we get ηB(xi) = 0 or ηB(xi) = 1.
Therefore Mα(B) = 0 if and only if when B is a crisp set.
b) To show the extremality condition that is to show Mα(B) is maximum if and only if
B is the most fuzzy set. Consider the earlier equation (16),

Mα(B) =
1

n(2α−1 − 1)

n∑
i=1

[ηB(xi)
2−α + (1− ηB(xi))

2−α − 1] (21)



V. M. Joshi, J. G. Dar / Eur. J. Pure Appl. Math, 17 (3) (2024), 2349-2360 2354

Table 1: Entropy Mα(B) at different values of α.

ηB(xi) Mα(B)
α = 0.5 α = 0.7 α = 1.1 α = 1.5

0 0 0 0 0

0.1 0.3911 0.4148 0.4936 0.6396

0.2 0.6658 0.6839 0.7381 0.8248

0.3 0.8536 0.8628 0.8889 0.9280

0.4 0.9637 0.9661 0.9729 0.9827

0.5 1.0000 1.0000 1.0000 1.0000

0.6 0.9637 0.9661 0.9729 0.9827

0.7 0.8536 0.8628 0.8889 0.9280

0.8 0.6658 0.6839 0.7381 0.8248

0.9 0.3911 0.4148 0.4936 0.6396

1.0 0 0 0 0

Figure 1: Entropy at different parametric values

differentiating it partially with respect to ηB(xi), we get,

∂Mα(B)

∂ηB(xi)
=

2− α

n(2α−1 − 1)

n∑
i=1

[ηB(xi)
1−α − (1− ηB(xi))

1−α] (22)

Case 1] When 0 ≤ ηB(xi) ≤ 0.5 and α < 0, α ̸= 1, α < 2 then
∂Mα(B)
∂ηB(xi)

is positive. (Refer Table 2)

Case 2] When 0.5 ≤ ηB(xi) ≤ 1 and α < 0, α ̸= 1, α < 2 then
∂Mα(B)
∂ηB(xi)

is negative.(Refer Table 2)

Case 3] When ηB(xi) = 0.5 that is, if B is a most Fuzzy set then

∂Mα(B)
∂ηB(xi)

= 2−α
n(2α−1−1)

∑n
i=1[(0.5)

1−α − (1− 0.5)1−α]

Therefore, the value of ∂Mα(B)
∂ηB(xi)

becomes zero. Hence Mα(B) is an increasing function of

ηB(xi) satisfying 0 ≤ ηB(xi) ≤ 0.5 and decreasing function for 0.5 ≤ ηB(xi) ≤ 1. Also at



V. M. Joshi, J. G. Dar / Eur. J. Pure Appl. Math, 17 (3) (2024), 2349-2360 2355

ηB(xi) = 0.5, the fuzzy entropy function vanishes. ThereforeMα(B) is a concave function
and has a global maximum at x = 0.5. Thus Mα(B) is maximum if and only if B is the
most fuzzy set.
c) Sharpness : Let B∗ be a sharpened version of B , i.e.
(i) If ηB(xi) < 0.5 , then η∗B(xi) ≤ ηB(xi)
(ii) If ηB(xi) > 0.5 , then η∗B(xi) ≥ ηB(xi)
Since Mα(B) is increasing function in the interval 0 ≤ ηB(xi) ≤ 0.5 and decreasing
function in the interval 0.5 ≤ ηB(xi) ≤ 1 thus
η∗B(xi) ≤ ηB(xi) =⇒Mα∗(B) ≤Mα(B) in [0, 0.5] and
η∗B(xi) ≥ ηB(xi) =⇒Mα∗(B) ≥Mα(B) in [0.5, 1]
Hence Mα∗(B) ≤Mα(B)
d) Symmetry: Since η(xi)

c = 1− η(xi) hence it is trivial to show that Mα(Bc) =Mα(B).
Hence all the four properties of fuzzy entropy measure are satisfied by Mα(B). Therefore
it is a valid fuzzy entropy measure.

Table 2: Partial derivative ∂Mα(B)
∂ηB(xi)

.

ηB(xi)
∂Mα(B)
∂ηB(xi)

α = 0.2 α = 0.5 α = 0.7 α = 0.9 α = 1.1 α = 1.3 α = 1.5 α = 1.9

0 4.2288 5.1213 6.9242 16.4260 ∞ ∞ ∞ ∞
0.1 3.2168 3.2390 3.2384 3.2062 3.1140 2.9168 0.2313 0.7902

0.2 2.3705 2.2903 2.2034 2.0794 1.9061 1.6699 0.1227 0.3504

0.3 1.5650 1.4797 1.3965 1.2877 1.1490 0.9755 0.0692 0.1821

0.4 0.7785 0.7280 0.6804 0.6202 0.5461 0.4566 0.0318 0.0805

0.5 0 0 0 0 0 0 0 0

0.6 −0.7785 −0.7280 −0.6804 −0.6202 −0.5461 −0.4566 −0.0318 −0.0805

0.7 −1.5650 −1.4797 −1.3965 −1.2877 −1.1490 −0.9755 −0.0692 −0.1821

0.8 −2.3705 −2.2903 −2.2034 −2.0794 −1.9061 −1.6699 −0.1227 −0.3504

0.9 −3.2168 −3.2390 −3.2384 −3.2062 −3.1140 −2.9168 −0.2313 −0.7902

1.0 −4.2288 −5.1213 −6.9242 −16.4260 −∞ −∞ −∞ −∞

Example 1. Let A be the fuzzy set defined as A = {(1, 0.2), (2, 0.8), (3, 0.5), (4, 0.7), (5, 0.3)}
and α = 0.2. Then the value of proposed generalized measure of entropy function Mα(A)
is

Mα(A) = 1
n(2α−1−1)

∑n
i=1[ηB(xi)

2−α + (1− ηB(xi))
2−α − 1]

Mα(A) = 1
5∗(20.2−1−1)

∑5
i=1([0.2

1.8+0.81.8−1]+ [0.81.8+0.21.8−1]+ [0.51.8+0.51.8−1]

+[0.71.8 + 0.31.8 − 1] + [0.31.8 + 0.71.8 − 1])
= 0.7966084

Example 2. Consider a set X = {2, 4, 7} and a fuzzy set B on X which is defined as
B = {(2, 0.4), (4, 0.6), (7, 0.1)}. Evaluate entropy function Mα(A) by taking α = 0.6.



V. M. Joshi, J. G. Dar / Eur. J. Pure Appl. Math, 17 (3) (2024), 2349-2360 2356

Figure 2: Partial derivative at different parametric values

Solution:
Mα(A) = 1

3∗(20.6−1−1)

∑3
i=1([0.4

1.4 + 0.61.4 − 1] + [0.61.4 + 0.41.4 − 1] + [0.11.4 + 0.91.4 − 1])

= 0.77720

4. Some properties of Mα(B) of order α

The proposed generalized measure of fuzzy entropy of order α has the following prop-
erties:

Theorem 2. For any two fuzzy sets R and S of universe of discourse X,
Mα(R ∪ S) +Mα(R ∩ S) =Mα(R) +Mα(S)

Proof. Let us divide the set X into two sets as :
X+={x/x ∈ X, ηA(xi) ≥ ηB(xi)}
X−={x/x ∈ X, ηA(xi) < ηB(xi)}

where ηR(xi) and ηS(xi) are the fuzzy membership values of R and S respectively. There-
fore,
Mα(R ∪ S) = 1

n(2α−1−1)

∑n
i=1[ηR∪S(xi)

2−α + (1− ηR∪S(xi))
2−α − 1]

using X+ the value of Mα(R ∪ S) becomes
Mα(R ∪ S) = 1

n(2α−1−1)

∑n
i=1[ηR(xi)

2−α + (1− ηR(xi))
2−α − 1]

also
Mα(R ∩ S)= 1

n(2α−1−1)

∑n
i=1[ηR∪S(xi)

2−α + (1− ηR∪S(xi))
2−α − 1]

using X− we get,
Mα(R ∩ S)= 1

n(2α−1−1)

∑n
i=1[ηR(xi)

2−α + (1− ηS(xi))
2−α − 1]

Mα(R ∪ S) +Mα(R ∩ S)
= 1

n(2α−1−1)

∑n
i=1[ηR∪S(xi)

2−α + (1− ηR∪S(xi))
2−α − 1]+

1
n(2α−1−1)

∑n
i=1[ηR∩S(xi)

2−α + (1− ηR∩S(xi))
2−α − 1]



V. M. Joshi, J. G. Dar / Eur. J. Pure Appl. Math, 17 (3) (2024), 2349-2360 2357

=Mα(R) +Mα(S)
Hence the result is proved.

Corollary 1. For any fuzzy set R in a universe of discourse X and Rc be the complement
of fuzzy set then
Mα(R) =Mα(R

c) =Mα(R ∪Rc) =Mα(R ∩Rc)

Proof. The proof is trivially follows from Theorem 2.

Theorem 3. For a fuzzy set R,S, T of set X, Mα(B) satisfies the following properties:
(i) Mα((R ∪ S) ∪ T ) =Mα(R ∪ (S ∪ T ))
(ii) Mα((R ∩ S) ∩ T ) =Mα(R ∩ (S ∩ T ))
(iii) Mα(R ∪ S) =Mα(R

c ∩ Sc)c

(iv) Mα(R ∩ S) =Mα(R
c ∪ Sc)c

Proof. Let Xp = {x/x ∈ X, ηR(xi) ≥ ηS(xi) ≥ ηT (xi)} and

Xq = {x/x ∈ X, ηR(xi) < ηS(xi) < ηT (xi)}

(i) To show Mα((R ∪ S) ∪ T ) =Mα(R ∪ (S ∪ T ))

Consider the left hand side:

Mα(R ∪ S)= 1
n(2α−1−1)

∑n
i=1[ηR∪S(xi)

2−α + (1− ηR∪S(xi))
2−α − 1]

= 1
n(2α−1−1)

∑n
i=1[ηR(xi)

2−α + (1− ηR(xi))
2−α − 1]

Mα((R ∪ S) ∪ T )
= 1

n(2α−1−1)

∑n
i=1[ηR∪T (xi)

2−α + (1− ηR∪T (xi))
2−α − 1]

= 1
n(2α−1−1)

∑n
i=1[ηR(xi)

2−α + (1− ηR(xi))
2−α − 1]

Now consider the right hand side:

Mα(R ∪ (S ∪ T ))

= 1
n(2α−1−1)

∑n
i=1[η(R∪(S∪T ))(xi)

2−α + (1− η(R∪(S∪T ))(xi))
2−α − 1]

= 1
n(2α−1−1)

∑n
i=1[ηR(xi)

2−α + (1− ηR(xi))
2−α − 1]

Hence Mα((R ∪ S) ∪ T ) =Mα(R ∪ (S ∪ T ))



V. M. Joshi, J. G. Dar / Eur. J. Pure Appl. Math, 17 (3) (2024), 2349-2360 2358

(ii) Similarly , associativity property holds for intersection also.

(iii) To show that Mα(R ∪ S) =Mα(R
c ∩ Sc)c

Consider Mα(R
c ∩ Sc)

= 1
n(2α−1−1)

∑n
i=1[ηRc∪Sc(xi)

2−α + (1− ηRc∪Sc(xi))
2−α − 1]

= 1
n(2α−1−1)

∑n
i=1[ηR∩S(xi)

2−α + (1− ηR∩S(xi))
2−α − 1]

Now Mα(R
c ∩ Sc)c

= 1
n(2α−1−1)

∑n
i=1[η

c
R∩S(xi)

2−α + (1− ηcR∩S(xi))
2−α − 1]

= 1
n(2α−1−1)

∑n
i=1[ηR∪S(xi)

2−α + (1− ηR∪S(xi))
2−α − 1]

=Mα(R ∪ S)
Hence proved. Exactly in the similar way property (iv) can be proved.

Theorem 4. Mα(B) attains the maximum when B is most fuzzy set and attains minimum
when B is least fuzzy set and it is independent of order α.

Proof. In Theorem no 1, it was already proved that Mα(B) is maximum if and only if
ηB(xi) = 0.5 that means B is most fuzzy set and minimum when B is a crisp set. Now to
prove that both these results are independent of α. Let B is most fuzzy set therefore put
µB(xi) = 0.5 in the following equation.
Mα(B) = 1

n(2α−1−1)

∑n
i=1[ηB(xi)

2−α + (1− ηB(xi))
2−α − 1]

= 1
n(2α−1−1)

∑n
i=1[0.5

2−α + (1− 0.5)2−α − 1]

= 1
n(2α−1−1)

∑n
i=1[0.5

2−α + (0.5)2−α − 1]

= 1
n(2α−1−1)

∑n
i=1[2

1
2

2−α − 1]

= 1
n(2α−1−1)

∑n
i=1[2

α−1 − 1]

= [2α−1−1]
n(2α−1−1)

∑n
i=1[1]

= 1.
which is independent of α.
On the other hand when B is least fuzzy set.That is B is a crisp set then ηB(xi) = 0 or
ηB(xi) = 1 then Mα(B) = 0 which is again independent of α.
Hence the theorem is proved.



REFERENCES 2359

5. Conclusion

In this paper, we have reviewed the concept of entropy in information theory for dis-
crete random variable and studied several generalizations of Shannon entropy. A brief
introduction about fuzzy sets and a journey from entropy to fuzzy entropy is discussed.
Numerical examples are provided for understanding the concept of proposed fuzzy en-
tropy measure. We have proposed a new parametric generalized fuzzy entropy measure
of Mathai-Haubold entropy and given the proof of validation. The particular cases have
been discussed in detail along with some of the properties of this fuzzy entropy measure.
For the future study, we will propose a new parametric generalizations of parametric fuzzy
entropy, a new divergence measures, total ambiguity and fuzzy improvement information
measures.

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