Generalized ‘Useful’ Converse Jensen’s Inequality with Data
Illustration
PANKAJ PRASAD DWIVEDI, DILIP KUMAR SHARMA
Department of Mathematics Jaypee University of Engineering and Technology, Guna, 473226 INDIA
Abstract: - In the present communication, we give the converse of generalized ‘useful’ Jensen inequality
and show that some recently reported inequalities are simple consequences of those results that have
been established for a long time. We also include a new improvement of the proposed inequality of
Jensen as well as changes to some associated outcomes, where generalized ‘useful’ converse of the
Inequality of Jensen is presented and implementations related to it are given in the theory of information.
Finally, it is shown with the help of numerical data that inequalities hold well both for convex and
concave functions.
Key-words: - Probability distribution; Jensen’s inequality; utility distribution; ‘useful’ Information
measure; Strongly convex function; ‘useful’ converse Jessen’s inequality.
Received: June 16, 2021. Revised: March 12, 2022. Accepted: April 14, 2022. Published: May 5, 2022.
1 Introduction
Let
be a set of all possible discrete
Chance distributions of a random variable
having utility distribution
attached to each
such that is the utility of an event
having the chance of occurrence .
Let be the set of positive
real numbers, where is the utility or importance
of the outcome . In general, the utility is
independent of the likelihood of encoding the
source symbol , that is .
The source of information is thus given by
(1)
Where
, is
called a utility information scheme. Corresponding
to the scheme (1) Belis and Guiasu [1] gave the
following measure of information:
(2)
The above measure (2) is called ‘useful
information’ and it reduces to Shannon's
information [14] when utilities are ignored, as seen
following:
(3)
By using various postulates, many authors
have defined the entropy of Shannon. Using
essential assumptions that were deduced by
Fadeev [7], Chandy and Mcliod [2], Kendall [9],
Khinchin [10] made Shannon's argument more
accurate. Tverberg [16], etc., was further defined
by the entropy of Shannon by considering various
sets of postulates. Simic [15] depicts a continuous
series of real numbers adhering to a defined finite
interval with a defined upper global bound, as well
as demonstrating with examples how this
technique can be used to establish the converse of
several key inequalities. For strongly convex and
strongly mid-convex functions, the counterparts of
the converse Jensen inequality were presented by
Klaricic & Nikodem [11]. Approximation theory,
mathematical economics, and optimization theory
all benefit from strong convex functions. Many of
their qualities have been documented in the
Literature for isotonic linear functionals Dragomir
[6] has given a reverse of Jensen’s inequality.
In several fields of mathematics, convex
functions play an essential role to Rashid et al. [13]
and Ge-Jile et al. [8]. They are particularly useful
in the study of optimization issues, where they
have a variety of useful features. The convex
function is an open set, for example, it contains just
one minimum. Convex functions continue to meet
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.7
Pankaj Prasad Dwivedi, Dilip Kumar Sharma
E-ISSN: 2224-2678
62
Volume 21, 2022
similar characteristics in infinite-dimensional
spaces given acceptable additional assumptions,
they are the most well-known basic aspect in the
calculus of variations as a result. The convex
enables control of the measured data of a random
variable that is always bounded above by the
convex function's expected value in probability
theory. Jensen's inequality, as well as Holder's
inequality and Arithmetic–Geometric mean
inequality, may be derived from this conclusion.
Convexity is something we encounter all the
time and in a variety of ways. The most typical
scenario is our standing stance, which is safe as
long as our center of gravity's vertical projection is
contained inside of the convex envelope of our
feet! Convexity also has a significant influence on
our daily lives due to its diverse uses in industry,
business, health, art, and other fields. Problems
with optimal resource allocation and non-
cooperative game equilibria are also present.
Because a convex function has a convex set as its
basis, the theory of convex functions falls within
the umbrella of convexity. Nonetheless, it is a
significant theory in and of itself, as it affects
practically all fields of mathematics.
The graphical analysis is most often the initial
issue that necessitates the acquaintance with this
theory. This is an opportunity to learn about the
second derivative proof of concavity, which is a
useful tool for detecting convexity. The difficulty
of identifying the extremal values of functions
with many variables, as well as the application of
Hessian as a higher dimensional generalization of
the second derivative, follows. The next step is to
go on to optimization issues in infinite-
dimensional spaces, however full of technological
complexity required to solve such issues, the
fundamental concepts are quite comparable to
those behind only one variable example.
We would like to highlight the introduction
and study of strongly convex functions, which play
a crucial contribution to information theory and
related fields. Many authors, for instance, strongly
convex functions were used to explaining the one-
of-a-kind presence of a possible answer to
nonlinear supplementary problems. In the study of
iterative approaches, the convergence towards
tackling variational inequalities and equilibrium
difficulties, strongly convex functions were also
critical. Using strongly convex functions,
Nikodem and Pales [12] explore the crucial
explanation of inner product spaces, which is an
innovative and unique application.
For convex functions, we obtained the
following converse of generalized ‘useful’
Inequality of Jensen's that reduce the inequality
given by S. S. Dragomir and N. M. Ionescu in [4]:
(4)
Suppose that is the interior of the interval
, and is differentiable convex on
, and
If on is strictly convex, then iff
the equality case holds in (4). The
above measure reduces to Dragomir [5], when
‘utilities are ignored. Several applications of this
can be found in Dragomir and Goh [3]. The key
contribution of this research is to highlight refining
of the converse of generalized ‘useful’ inequality
of Jensen's defined in (4).
2 New Improvements
In this section, we have given some lemma
and their proofs where utilities are attached to
probabilities of a differentially convex function
and differentially strictly convex function and
basic results that will be needed in this
correspondence.
Lemma 2.1 Suppose a differentiable convex
function on defined as and
are the utilities attached to probabilities and
with
,
then we have the inequality
(5)
Proof. The following inequality hold for all
, if is differentiable convex on :
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.7
Pankaj Prasad Dwivedi, Dilip Kumar Sharma
E-ISSN: 2224-2678
63
Volume 21, 2022
(6)
Again, we get the next inequality if we multiply
with and choose in (6),
, and sum over from 1 up to .
it is equal to the following inequality
(7)
For each
We deduce (5), taking the infimum over .
The following outcome relates to the refining of
Dragomir-Ionescu (4). It can be noted that (5)
reduces the inequality given by Dragomir [5] when
utilities are ignored.
Theorem 2.1 Suppose a differentiable convex
function on defined as , and
are the utilities attached to probabilities and
with
, then
(8)
Proof. In the above inequality (8) the second
inequality is followed by the first inequality in (5).
It's indeed obvious that
where
and therefore the last
portion of (8) is then proven.
We may utilize the following result for
applications.
Lemma 2.2 Suppose a differentiable strictly
convex function on defined as ,
and are the utilities attached to probabilities
and with
, then
(9)
where represents the opposite function of
the derivative defined on If
, then equality case holds in (9).
Proof. Define the function
. Consequently, is
differentiable on and then
(10)
If and then the above equation
is equivalent to as follows
(11)
and since
is one-to-one,
existence strictly increasing on , The equation
(11) therefore has a unique solution
alternatively to given by
(12)
The derivative of defined on , where
is the inverse function of the derivative.
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.7
Pankaj Prasad Dwivedi, Dilip Kumar Sharma
E-ISSN: 2224-2678
64
Volume 21, 2022
Let, if and if
, then it follows that
By using (4), we deduce (9). It is clear that (9)
reduces the inequality given by Dragomir [5] when
utilities are ignored. The equality case assumes the
strict convexity of and specifics are omitted. It
is now feasible to disclose the next step in the
Dragomir-Ionescu process (4).
Theorem 2.2 Suppose a differentiable strictly
convex function on defined as ,
and are the utilities attached to probabilities
and with
, then
(13)
Iff , the equality holds in (13).
Proof. By lemma 2.1 and theorem 2.1, the proof is
obvious.
Remark 2.1 In Lemma 2.2, we note that there is
double inequality with the assumptions
(14)
with equality iff
In case, if is a strictly concave and differentiable
function, then
(15)
The proof of (15) follows by (14) choosing
, with equality iff .
3 Numerical and Graphical Illustration
In this section, we give a numerical result that
will further strengthen our results (14) and (15).
Suppose a differentiable strictly convex function
on defined as ,
and let
are the utilities that are attached to
probabilities
with
. Now we take the convex
function is an example of one. The
'useful' Jensen's inequality asserts that we must
discover the value of the 'useful Jensen inequality
for convex function
in Table 1, we measure the numerical value for
taking examples of the convex function, the value
of
and the value of
which holds the inequality
(14) and (15). A bar graph for the convex function
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.7
Pankaj Prasad Dwivedi, Dilip Kumar Sharma
E-ISSN: 2224-2678
65
Volume 21, 2022
is shown in Figure 1. If we choose any two
locations on the graph of a function and draw a line
segment between them, the entire segment is
above the graph, then the function is convex. The
function, on the other hand, is considered to be
concave if the line segment always sits below the
graph. To put it another way, is convex only,
if is concave.
Table 1. The function evaluated at the expectation
1
1
1/6
2
0.17
0.17
0.03
0.03
2
4
1/6
3
0.33
0.67
0.06
0.11
3
9
1/6
4
0.50
1.50
0.08
0.25
4
16
1/6
5
0.67
2.67
0.11
0.44
5
25
1/6
6
0.83
4.17
0.14
0.69
6
36
1/6
7
1.00
6.00
0.17
1.00
=0.34
=2.53
Figure 1. Bar graph representation of inequality for the convex function.
4 Discussion
In this paper, we have mainly worked on a
differentiable convex function in which a utility
test is done as well as we have displayed the result
in terms of a differentiable function and its related
result also, we have shown the result in terms of
Jensen’s inequalities. Belis and Guiasu gave a
measure of information for the discrete probability
distribution of random variables that are required
for information theory.
5 Conclusion
This paper introduced the converse of generalized
‘useful’ Jensen inequality. Our work reinforces
Dragomir's fundamental result through a stronger
and more generalized 'useful' inequality of Jensen,
which can be used further in information theory.
With the help of numerical data, it is shown that
inequality holds for both convex and concave
functions.
0
5
10
15
20
25
30
35
40
𝑦 𝑦^2 𝑔[𝐸(𝑦)] 𝐸[𝑔(𝑦)]
Series1 Series2 Series3 Series4 Series5 Series6
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.7
Pankaj Prasad Dwivedi, Dilip Kumar Sharma
E-ISSN: 2224-2678
66
Volume 21, 2022
References:
[1] Belis, M. and S. Guiasu. (1968). A
quantitative-qualitative measure of
information in Cybernetics System, IEEE
Trans. Inform. Theory IT 14: 593–594.
[2] Chandy, T.W. And J.B. Mcliod (1960). On
a functional equation, Proc. Edinburgh
Math’s 43: 7–8.
[3] Dragomir, S. S., C. J. Goh (1996). A
counterpart of Jensen’s discrete inequality
for differentiable convex mappings and
applications in information theory, Math.
Comput. Modelling 24(2): 1–11.
[4] Dragomir, S. S., N. M. Ionescu (1994).
Some converse of Jensen’s inequality and
applications, Anal. Num. Theor. Approx
23: 71–78.
[5] Dragomir, S.S. (2001, a). On a converse of
Jensen’s Inequality, Univ. Beograd. Publ.
Elektrotehn. Fak.Ser. Mat. 12: 48–51.
[6] Dragomir, S.S. (2001, b), on a reverse of
Jensen’s inequality for isotonic linear
functionals, journal of Inequalities in Pure
and Applied Mathematics, volume 2, issue
3, article 36.
[7] Fadeev, D.K. (1956). On the concept of
entropies of finite probabilistic scheme,
(Russian) Uspchi Math. Nauk 11: 227–
231.
[8] JiLe G., H. Rashid., Farooq F., and
Sultana, S. (2021). Some Inequalities for a
New Class of Convex Functions with
Applications via Local Fractional Integral,
Journal of Function Spaces, vol. 2021,
https://doi.org/10.1155/2021/6663971.
[9] Kendall, D.G. (1964). Functional
equations in information theory, Z.Wahrs
Verw.Geb 2: 225–229.
[10] Khinchin, A.I. (1957). Mathematical
Foundations of Information Theory,
Dover Publications New York.
[11] Klaricic Bakula, M., & Nikodem, K.
(2016). On the converse Jensen inequality
for strongly convex functions. Journal of
Mathematical Analysis and Applications,
434(1):516–522. https://doi:
10.1016/j.jmaa.2015.09.032.
[12] Nikodem, K.; Pales, Z.S. (2011).
Characterizations of inner product spaces
by strongly convex functions. Banach J.
Math. Anal., vol. 1: 83–87.
[13] Rashid S., T. Abdeljawad, F. Jarad, and
M. A. Noor (2019). Some estimates for
generalized Riemann-Liouville fractional
integrals of exponentially convex
functions and their
applications, Mathematics, vol.7(9): 807.
[14] Shannon, C.E. (1948). A mathematical
theory of communication, Bell System
Technical Journal 27: 379–423(Part I):
623–656(Part II).
[15] Simic, S. (2009). On a Converse of
Jensen’s Discrete Inequality. Journal of
Inequalities and Applications, (1),
153080. doi:10.1155/2009/153080.
[16] Tverberg, H. (1958). A new derivation of
the information function, Math. Scand 6:
297–298.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the Creative
Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en_US
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.7
Pankaj Prasad Dwivedi, Dilip Kumar Sharma
E-ISSN: 2224-2678
67
Volume 21, 2022
