Several Jensen–Grüss inequalities with applications in information theory
UDC 517.5 Several integral Jensen–Grüss  inequalities are proved together with their refinements.  Some new bounds for integral Jensen–Chebyshev  inequality are obtained. The multidimensional integral variants are also pre...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512438202400768 |
|---|---|
| author | Butt, S. I. Pečarić, Ð. Pečarić, J. Butt, S. I. Pečarić, Ð. Pečarić, J. |
| author_facet | Butt, S. I. Pečarić, Ð. Pečarić, J. Butt, S. I. Pečarić, Ð. Pečarić, J. |
| author_sort | Butt, S. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-01-23T14:02:43Z |
| description | UDC 517.5
Several integral Jensen–Grüss  inequalities are proved together with their refinements.  Some new bounds for integral Jensen–Chebyshev  inequality are obtained. The multidimensional integral variants are also presented.  In addition, some integral Jensen–Grüss  inequalities for monotone  and completely monotone functions are established.  Finally, as an application, we present the refinements  for Shannon's entropy. |
| doi_str_mv | 10.37863/umzh.v74i12.6554 |
| first_indexed | 2026-03-24T03:28:47Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i12.6554
UDC 517.5
S. I. Butt1 (COMSATS Univ. Islamabad, Lahore Campus, Pakistan),
D. Pečarić, J. Pečarić (Catholic Univ. Croatia, Zagreb)
SEVERAL JENSEN – GRÜSS INEQUALITIES
WITH APPLICATIONS IN INFORMATION THEORY
КIЛЬКА НЕРIВНОСТЕЙ ЙЄНСЕНА – ГРЮССА
ТА ЇХ ЗАСТОСУВАННЯ В ТЕОРIЇ IНФОРМАЦIЇ
Several integral Jensen – Grüss inequalities are proved together with their refinements. Some new bounds for integral Jensen –
Chebyshev inequality are obtained. The multidimensional integral variants are also presented. In addition, some integral
Jensen – Grüss inequalities for monotone and completely monotone functions are established. Finally, as an application, we
present the refinements for Shannon’s entropy.
Доведено кiлька iнтегральних нерiвностей Йєнсена – Грюсса та їх уточнення. Отримано деякi новi оцiнки для
iнтегральної нерiвностi Йєнсена – Чебишова. Також наведено багатовимiрнi iнтегральнi варiанти. Крiм того, вста-
новлено деякi iнтегральнi нерiвностi Йєнсена – Грюсса для монотонних i цiлком монотонних функцiй. Насамкiнець
в якостi додатка наведено уточнення, отриманi для ентропiї Шеннона.
1. Introduction. Jensen inequality is the most notable inequality and many other inequalities can be
deduced from it as its consequences. This inequality has huge impact in solving many optimization
problems, e.g., information theory, probability theory, applied statistics, control theory and computer
sciences. Taking into consideration the tremendous applications of Jensen’s inequality in various
fields of mathematics and other applied sciences, the generalizations and improvements of Jensen’s
inequality has been a topic of supreme interest for the researchers during the last few decades as
evident from a large number of publications on the topic see [4, 7, 10, 11, 19, 21].
Theorem A (classical Jensen’s inequality, see [18]). Let h be an integrable function on a pro-
bability space (\Omega ,\scrA , \mu ) taking values in an interval l \subset \BbbR . Then
\int
\Omega
hd\mu lies in I. If \varphi is a convex
function on I such that \varphi \circ h is integrable, then
\varphi
\left( \int
\Omega
hd\mu
\right) \leq
\int
\Omega
\varphi \circ hd\mu .
There are two other important inequalities in mathematical analysis namely Chebyshev inequality
[18, p. 197] or [12, p. 240] and Grüss inequality [8]. To start with, we let \varphi , h \in L[u, v] and \rho :
[u, v] \rightarrow \BbbR + be Lebesgue integrable functions. Then we consider the following weighted Chebyshev
functional:
\frakC (\varphi , h; \rho ) =
1
P
v\int
u
\rho (\zeta )\varphi (\zeta )h(\zeta )d\zeta - 1
P
v\int
u
\rho (\zeta )\varphi (\zeta )d\zeta
1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta , (1)
where P =
\int v
u
\rho (\zeta )d\zeta .
1 Corresponding author, e-mail: saadihsanbutt@gmail.com.
c\bigcirc S. I. BUTT, D. PEČARIĆ, J. PEČARIĆ, 2022
1654 ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
SEVERAL JENSEN – GRÜSS INEQUALITIES WITH APPLICATIONS IN INFORMATION THEORY 1655
If \rho (\zeta ) = 1 for all \zeta \in [u, v], then we define Chebyshev functional \frakC (\varphi , h) = \frakC (\varphi , h; 1).
The renowned Grüss inequality states that
\frakC (\varphi , h; \rho ) \leq 1
4
(\Lambda - \lambda )(\Psi - \psi ), (2)
where \Lambda , \lambda ,\Psi , \psi are real numbers with the property
- \infty < \lambda \leq \varphi \leq \Lambda <\infty , - \infty < \psi \leq h \leq \Psi <\infty a.e. on [u, v]. (3)
In 1934, G. Grüss [8] gives proof without weights however the same proof hold for weighted version
also.
Moreover, we need to mention here the weighted version of Korkine’s identity [12, p. 242]
\frakC (\varphi , h; \rho ) =
1
2P 2
v\int
u
v\int
u
\rho (\tau )\rho (\zeta )(\varphi (\tau ) - \varphi (\zeta ))(h(\tau ) - h(\zeta ))d\tau d\zeta . (4)
We give variety of upper bounds for Jensen’s difference in terms of the Grüss and Chebyshev
inequalities. We also present multidimensional case of Jensen – Grüss inequality and formulate its
bounds in case for monotonic functions. We also point out some applications of such results in
information theory, namely we provide some new upper bounds for the Shannon entropy.
2. Jensen – Grüss inequality.
Theorem 2.1. Let \varphi : I = [u, v] \subset \BbbR \rightarrow \BbbR be differentiable mapping with continuous first
derivative. Let h : I \rightarrow I such that h, \varphi \circ h, \varphi \prime \circ h \in L[u, v], and suppose that there exist
\lambda ,\Lambda , \psi ,\Psi \in \BbbR such that
\lambda \leq h(\zeta ) \leq \Lambda , \psi \leq \varphi \prime (\zeta ) \leq \Psi for all \zeta \in I.
Then, for all \rho (\zeta ) \geq 0 such that P =
\int v
u
\rho (\zeta )d\zeta > 0 exists, we have the following refinements:
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1P
v\int
u
\rho (\zeta )(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \Psi - \psi
2
\left( 1
P
v\int
u
\rho (\zeta )h2(\zeta )d\zeta -
\left( 1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta
\right) 2\right)
1
2
\leq
\leq (\Lambda - \lambda )(\Psi - \psi )
4
. (5)
Proof. Employing the mean-value theorem for points c, d \in I, we can write that there exists \xi ,
c \leq \xi \leq d, such that
\varphi (c) - \varphi (d) = \varphi \prime (\xi )(c - d). (6)
Using (6) for
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1656 S. I. BUTT, D. PEČARIĆ, J. PEČARIĆ
c = \=h =
1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta
and d = h, we conclude that there exists g, \=h \leq g \leq h, such that
\varphi
\bigl(
\=h
\bigr)
- \varphi (h) = \varphi \prime (g)
\bigl(
\=h - h
\bigr)
. (7)
Now multiplying (7) by \rho (\zeta ) and integrating over [u, v] yields
P\varphi (\=h) -
v\int
u
\rho (\zeta )\varphi (h(\zeta ))d\zeta = \=h
v\int
u
\rho (\zeta )\varphi \prime (g(\zeta ))d\zeta -
v\int
u
\rho (\zeta )\varphi \prime (g(\zeta ))h(\zeta )d\zeta .
Dividing by P, we get
1
P
v\int
u
\rho (\zeta )(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta
\right) =
=
1
P
v\int
u
\rho (\zeta )\varphi \prime (g(\zeta ))h(\zeta )d\zeta - 1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta
1
P
v\int
u
\rho (\zeta )\varphi \prime (g(\zeta ))d\zeta .
Now taking modulus on both sides and using weighted Korkine’s identity (4), gives\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1P
v\int
u
\rho (\zeta )(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1P
v\int
u
\rho (\zeta )\varphi \prime (g(\zeta ))h(\zeta )d\zeta - 1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta
1
P
v\int
u
\rho (\zeta )\varphi \prime (g(\zeta ))d\zeta
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \frakC (h, \varphi \prime (g); \rho )
\bigm| \bigm| \leq 1
2P 2
v\int
u
v\int
u
\rho (\zeta )\rho (\tau )(| h(\zeta ) - h(\tau )| )\times
\times
\bigl( \bigm| \bigm| \varphi \prime (g(\zeta )) - \varphi \prime (g(\tau ))
\bigm| \bigm| \bigr) d\zeta d\tau .
Now applying Cauchy – Buniakowsky – Schwartz inequality for double integrals, we can state that
the last expression is less than\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1P
v\int
u
\rho (\zeta )(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \frakC
1
2 (h, h; \rho )\frakC
1
2
\bigl(
\varphi \prime (g), \varphi \prime (g); \rho
\bigr)
. (8)
Now utilizing weighted Grüss inequality (2) on second term, we obtain
\frakC
1
2 (h, h; \rho )
1
2
(\Psi - \psi ) =
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
SEVERAL JENSEN – GRÜSS INEQUALITIES WITH APPLICATIONS IN INFORMATION THEORY 1657
=
\left( 1
P
v\int
u
\rho (\zeta )h2(\zeta )d\zeta -
\left( 1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta
\right) 2\right)
1
2
\Psi - \psi
2
.
Now utilizing weighted Grüss inequality (2) on first term, we get
\leq (\Lambda - \lambda )(\Psi - \psi )
4
.
Theorem 2.1 is proved.
Remark 2.1. It is important to note that the first inequality in (5) is valid without the bounds of
function h.
We give the following interesting corollaries.
Corollary 2.1. Under the assumptions of Theorem 2.1, suppose that \varphi \prime is Lipschitzian with the
constant L > 0, i.e., \bigm| \bigm| \varphi \prime (x) - \varphi \prime (y)
\bigm| \bigm| \leq L| x - y|
for all x, y \in \mathrm{R}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}(h), where \lambda \leq \mathrm{R}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}(h) \leq \Lambda . Then we have the following refinements:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1P
v\int
u
\rho (\zeta )(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq L
\Lambda - \lambda
2
\left( 1
P
v\int
u
\rho (\zeta )h2(\zeta )d\zeta -
\left( 1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta
\right) 2\right)
1
2
\leq
\leq L
(\Lambda - \lambda )2
4
.
Proof. By using (8) and Lipschitzian condition on \varphi \prime , we get\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1P
v\int
u
\rho (\zeta )(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq L\frakC
1
2 (h, h; \rho )\frakC
1
2 (id(\zeta ), id(\zeta ); \rho ),
where id(\zeta ) = \zeta . Now applying weighted Grüss inequality (2) successively on right-hand side, we
obtain
\leq L
\Lambda - \lambda
2
\frakC
1
2 (h, h; \rho ) \leq
\leq L
(\Lambda - \lambda )2
4
.
Corollary 2.2. With the assumptions of Corollary 2.1, further suppose that \varphi \prime \prime is bounded, that
is, L = \| \varphi \prime \prime \| and \| \cdot \| is defined as the sup-norm. Then we get the following refinements:
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1658 S. I. BUTT, D. PEČARIĆ, J. PEČARIĆ\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1P
v\int
u
\rho (\zeta )(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm\| \bigm\| \varphi \prime \prime \bigm\| \bigm\| \Lambda - \lambda
2
\left( 1
P
v\int
u
\rho (\zeta )h2(\zeta )d\zeta -
\left( 1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta
\right) 2\right)
1
2
\leq
\leq \| \varphi \prime \prime \| (\Lambda - \lambda )2
4
.
3. Jensen – Chebyshev inequality. We need the following lemma of our interest.
Lemma 3.1 [3]. Let h : [u, v] \rightarrow \BbbR be an absolutely continuous function such that (h\prime )2 \in
\in L[u, v] and weight \rho be a positive integrable function such that
P (z) =
z\int
u
\rho (\zeta )d\zeta and \u P (z) = P (z)
v\int
u
\zeta \rho (\zeta )d\zeta - P
z\int
u
\zeta \rho (\zeta )d\zeta .
Then we have the following inequality:
\frakC (h, h; \rho ) \leq 1
P 2
v\int
u
\u P (z)[h\prime (z)]2dz (9)
provided that integral on the right-hand side of above inequality exists. Also the inequality in (9) is
sharp.
Theorem 3.1. Let \varphi : I = [u, v] \subset \BbbR \rightarrow \BbbR be differentiable mapping with continuous first
derivative. Let h : I \rightarrow I be absolutely continuous such that h, \varphi \circ h, \varphi \prime \circ h, (h\prime )2 \in L[u, v], and
suppose that there exist \psi , \Psi \in \BbbR such that
\psi \leq \varphi \prime (\zeta ) \leq \Psi for all \zeta \in I.
Then, for all \rho \geq 0 such that P =
\int v
u
\rho (\zeta )d\zeta > 0 exists and \u P (\cdot ) be as given in Lemma 3.1, we
have the following refinements:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1P
v\int
u
p(\zeta )(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1
P
v\int
u
p(\zeta )h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \Psi - \psi
2
\left( 1
P
v\int
u
p(\zeta )h2(\zeta )d\zeta -
\left( 1
P
v\int
u
p(\zeta )h(\zeta )d\zeta
\right) 2\right)
1
2
\leq
\leq \Psi - \psi
2P
\left( v\int
u
\u P (\zeta )
\bigl[
h\prime (\zeta )
\bigr] 2
d\zeta
\right) 1
2
. (10)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
SEVERAL JENSEN – GRÜSS INEQUALITIES WITH APPLICATIONS IN INFORMATION THEORY 1659
Proof. We have already established in Theorem 2.1 that\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1P
v\int
u
\rho (\zeta )(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1P
v\int
u
p(\zeta )\varphi \prime (g(\zeta ))h(\zeta )d\zeta - 1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta
1
P
v\int
u
\rho (\zeta )\varphi \prime (g(\zeta ))d\zeta
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \frakC (h, \varphi \prime (g); \rho )
\bigm| \bigm| \leq \frakC
1
2 (h, h; \rho )\frakC
1
2
\bigl(
\varphi \prime (g), \varphi \prime (g); \rho
\bigr)
.
Now utilizing weighted Grüss inequality (2) on second term, we get
\leq \frakC
1
2 (h, h; \rho )
1
2
(\Psi - \psi ) =
=
\left( 1
P
v\int
u
\rho (\zeta )h2(\zeta )d\zeta -
\left( 1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta
\right) 2\right)
1
2
\Psi - \psi
2
.
Now utilizing Lemma 3.1 on first term, we obtain
\leq
\left( 1
P 2
v\int
u
\u P (\zeta )
\bigl[
h\prime (\zeta )
\bigr] 2
d\zeta
\right) 1
2
\Psi - \psi
2
.
Theorem 3.1 is proved.
4. Jensen – Chebyshev norm estimates. We start this section by notating the following classes
that we used:
(M1) C(u, v) denote the space of all functions \rho > 0 continuous on (u, v) such that
v\int
u
\rho (\zeta )d\zeta = P <\infty .
(M2) W
2
r (u, v) denote the space of all functions h which are locally absolutely continuous on
(u, v), with
v\int
u
rh\prime 2(\zeta )d\zeta <\infty .
Define
\| h\| r =
\left( v\int
u
r(\zeta )h2(\zeta )d\zeta
\right) 1
2
.
In [13], G. V. Milovanović and I. Z. Milovanović gave weighted norm estimates of Chebyshev
functional.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1660 S. I. BUTT, D. PEČARIĆ, J. PEČARIĆ
Theorem 4.1. Let \rho \in C(u, v), r(\zeta ) =
1
\rho (\zeta )
and h, g \in W 2
r (u, v). Then the following inequa-
lity holds:
| \frakC (h, g; \rho )| \leq P
\pi 2
\bigm\| \bigm\| h\prime \bigm\| \bigm\|
r
\bigm\| \bigm\| g\prime \bigm\| \bigm\|
r
. (11)
If h(\zeta ) = A+B \mathrm{s}\mathrm{i}\mathrm{n}\Delta (\zeta ), g(\zeta ) = C +D \mathrm{s}\mathrm{i}\mathrm{n}\Delta (\zeta ), where
\Delta (\zeta ) =
\left( \pi
P
b\int
\zeta
\rho (t)dt -
\zeta \int
a
\rho (t)dt
\right) ,
the equality appears in (11).
Now we are in position to state our results of this section.
Theorem 4.2. Let \varphi : I = (u, v) \subset \BbbR \rightarrow \BbbR be differentiable mapping with continuous first
derivative. Let h : I \rightarrow I be such that h \in W 2
r (I) and h, \varphi \circ h, \varphi \prime \circ h \in L[u, v] and suppose that
there exist \psi , \Psi \in \BbbR such that
\psi \leq \varphi \prime (\zeta ) \leq \Psi for all \zeta \in I.
Then, for all \rho \in C(u, v), r(\zeta ) =
1
\rho (\zeta )
, we have the following refinements:
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1P
v\int
u
p(\zeta )(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1
P
v\int
u
p(\zeta )h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \Psi - \psi
2
\left( 1
P
v\int
u
p(\zeta )h2(\zeta )d\zeta -
\left( 1
P
v\int
u
p(\zeta )h(\zeta )d\zeta
\right) 2\right)
1
2
\leq
\leq \Psi - \psi
2
\surd
P
\pi
\bigm| \bigm| h\prime \bigm| \bigm|
r
. (12)
Proof. We have already established in Theorem 2.1 that\bigm| \bigm| \frakC \bigl( h, \varphi \prime (g); \rho
\bigr) \bigm| \bigm| \leq \frakC
1
2 (h, h; \rho )\frakC
1
2
\bigl(
\varphi \prime (g), \varphi \prime (g); \rho
\bigr)
\leq \frakC
1
2 (h, h; \rho )
1
2
(\Psi - \psi ) =
=
\left( 1
P
v\int
u
\rho (\zeta )h2(\zeta )d\zeta -
\left( 1
P
v\int
u
\rho (\zeta )h(\zeta )d\zeta
\right) 2\right)
1
2
\Psi - \psi
2
.
Now utilizing Theorem 4.1 on first term, we get
\leq
\biggl(
P
\pi 2
\| h\prime \| 2r
\biggr) 1
2 \Psi - \psi
2
.
Theorem 4.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
SEVERAL JENSEN – GRÜSS INEQUALITIES WITH APPLICATIONS IN INFORMATION THEORY 1661
5. Multidimensional Jensen – Grüss inequality. Let (\Omega ,\scrA , \mu ) be a space with positive finite
measure. Let L = L1(\Omega ,\scrA , \mu ) and for h \in L define
\=h =
1
\mu (\Omega )
\int
\Omega
h(\zeta )d\mu (\zeta ).
Let U \subset \BbbR n be a convex set, and \varphi be an arbitrary convex function on U. If h1, h2, . . . , hn are
functions in class L1 (i.e., \mu -measurable functions), then the following multidimensional version of
Jensen’s integral inequality [18, p. 51] is valid:
1
\mu (\Omega )
\int
\Omega
\varphi (h1(\zeta ), h2(\zeta ), . . . , hn(\zeta ))d\mu (\zeta ) -
- \varphi
\left( 1
\mu (\Omega )
\int
\Omega
h1(\zeta )d\mu (\zeta ),
1
\mu (\Omega )
\int
\Omega
h2(\zeta )d\mu (\zeta ), . . . ,
1
\mu (\Omega )
\int
\Omega
hn(\zeta )d\mu (\zeta )
\right) \geq 0.
Inorder to give multidimensional Jensen – Grüss integral version, we denote
\bfh (\zeta ) = (h1(\zeta ), h2(\zeta ), . . . , hn(\zeta ))
be n-tuple of functions of class L1 and we denote
1
\mu (\Omega )
\int
\Omega
\bfh (\zeta )d\mu (\zeta )
be the n-tuple\left( 1
\mu (\Omega )
\int
\Omega
h1(\zeta )d\mu (\zeta ),
1
\mu (\Omega )
\int
\Omega
h2(\zeta )d\mu (\zeta ), . . . ,
1
\mu (\Omega )
\int
\Omega
hn(\zeta )d\mu (\zeta )
\right) .
Theorem 5.1. Let \varphi : U \subset \BbbR n \rightarrow \BbbR be differentiable mapping with continuous partial deriva-
tives, where U is a convex point set in \BbbR n. Let (\Omega ,\scrA , \mu ) be a space with positive finite measure and
L = L1(\Omega ,\scrA , \mu ). Also, let \bfh : \Omega \rightarrow U \subset \BbbR n such that hi(\zeta ), \varphi \circ \bfh (\zeta ), \varphi \prime \circ \bfh (\zeta ) \in L1 (i.e., \mu -
measurable functions) for all \zeta \in \Omega and i = 1, 2, . . . , n. Suppose that there exist \bfitlambda ,\bfLambda ,\bfitpsi ,\bfPsi \in \BbbR n
such that
\bfitlambda \leq \bfh \leq \bfLambda (the order is onsiderd cordinatewise)
and
\bfitpsi \leq \nabla \varphi (\bfitzeta ) \leq \bfPsi for all \bfitzeta \in \mathrm{d}\mathrm{o}\mathrm{m}(\varphi ).
Then we have the inequalities\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1
\mu (\Omega )
\int
\Omega
\varphi (h1(\zeta ), h2(\zeta ), . . . , hn(\zeta ))d\mu (\zeta ) -
- \varphi
\left( 1
\mu (\Omega )
\int
\Omega
h1(\zeta )d\mu (\zeta ),
1
\mu (\Omega )
\int
\Omega
h2(\zeta )d\mu (\zeta ), . . . ,
1
\mu (\Omega )
\int
\Omega
hn(\zeta )d\mu (\zeta )
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
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1662 S. I. BUTT, D. PEČARIĆ, J. PEČARIĆ
\leq 1
2
\| \bfPsi - \bfitpsi \|
\left( 1
\mu (\Omega )
\int
\Omega
\| \bfh (\zeta )\| 2d\mu (\zeta ) -
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| 1
\mu (\Omega )
\int
\Omega
\bfh (\zeta )d\mu (\zeta )
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
2\right)
1
2
\leq
\leq 1
4
\| \bfLambda - \bfitlambda \| \| \bfPsi - \bfitpsi \| .
Proof. Employing the mean-value theorem in multidimensional case for points \bfc ,\bfd \in \mathrm{d}\mathrm{o}\mathrm{m}(\varphi ),
we conclude that there exist \alpha \in (0, 1) such that
\varphi (\bfc ) - \varphi (\bfd ) = \langle \nabla \varphi (\bfitxi ), \bfc - \bfd \rangle , (13)
where \bfitxi = \bfd + \alpha (\bfc - \bfd ). Using (13) for
\bfc = \=\bfh =
\bigl(
\=h1, \=h2, . . . , \=hn
\bigr)
=
1
\mu (\Omega )
\int
\Omega
\bfh (\zeta )d\mu (\zeta )
be the n-tuple\left( 1
\mu (\Omega )
\int
\Omega
h1(\zeta )d\mu (\zeta ),
1
\mu (\Omega )
\int
\Omega
h2(\zeta )d\mu (\zeta ), . . . ,
1
\mu (\Omega )
\int
\Omega
hn(\zeta )d\mu (\zeta )
\right) ,
\bfd = \bfh = (h1, h2, . . . , hn) and \bfitxi = \bfg = (g1, g2, . . . , gn), where gi(\zeta ) \in L1 (i.e., \mu -measurable
functions) for all \zeta \in \Omega and i = 1, 2, . . . , n, we have
\varphi
\bigl(
\=\bfh
\bigr)
- \varphi (\bfh ) =
\bigl\langle
\nabla \varphi (\bfg ), \=\bfh - \bfh
\bigr\rangle
.
Integrating over \Omega w.r.t. \mu yields
\mu (\Omega )\varphi
\bigl(
\=\bfh
\bigr)
-
\int
\Omega
\varphi (\bfh (\zeta ))d\mu (\zeta ) =
\int
\Omega
\bigl\langle
\nabla \varphi (\bfg ), \=\bfh
\bigr\rangle
d\mu (\zeta ) -
\int
\Omega
\langle \nabla \varphi (\bfg ),\bfh \rangle d\mu (\zeta ).
Dividing by \mu (\Omega ), we get
1
\mu (\Omega )
\int
\Omega
\varphi (h1(\zeta ), h2(\zeta ), . . . , hn(\zeta ))d\mu (\zeta ) -
- \varphi
\left( 1
\mu (\Omega )
\int
\Omega
h1(\zeta )d\mu (\zeta ), . . . ,
1
\mu (\Omega )
\int
\Omega
hn(\zeta )d\mu (\zeta )
\right) =
=
1
\mu (\Omega )
\int
\Omega
\langle \nabla \varphi (\bfg ),\bfh \rangle d\mu (\zeta ) -
\Biggl\langle
1
\mu (\Omega )
\int
\Omega
\nabla \varphi (\bfg (\zeta ))d\zeta , \=\bfh
\Biggr\rangle
.
Rest of the proof can be completed by method used to prove multidimensional discrete version, given
in the proof of Theorem 1 from [6].
A multidimensional generalization of Lupas – Ostrowski inequality was given in [20]. For in-
stance, we give the following theorem for two variables.
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SEVERAL JENSEN – GRÜSS INEQUALITIES WITH APPLICATIONS IN INFORMATION THEORY 1663
Theorem 5.2 [20]. Let
\rho \in C(u1, v1), q \in C(u2, v2), \rho > 0, q > 0,
v1\int
u1
\rho (t)dt = P <\infty ,
v2\int
u2
q(t)dt = Q <\infty .
Let h : (u1, v1) \times (u2, v2) \rightarrow \BbbR be a function such that h(\cdot , x2) is locally absolutely continuous on
(u1, v1) for almost every x2 \in (u2, v2) and h(x1, \cdot ) is locally absolutely continuous on (u2, v2) for
almost every x1 \in (u1, v1). Suppose that
v1\int
u1
v2\int
u2
\rho (x1)q(x2)h
2(x1, x2)dx1dx2 <\infty
and
v1\int
u1
v2\int
u2
\Biggl[
q(x2)
\rho (x1)
\biggl(
\partial h
\partial x1
\biggr) 2
+
\rho (x1)
q(x2)
\biggl(
\partial h
\partial x1
\biggr) 2
\Biggr]
dx1dx2 <\infty .
Also let g satisfy the same condition as h, then we have
| \frakC (h, g; \rho , q)| \leq 1
\pi 2
\| \nabla h; \rho , q\| 2\| \nabla g; \rho , q\| 2,
where
\frakC (h, g; \rho , q) = A(h, g; \rho , q) - A(h; \rho , q)A(g; \rho , q), (14)
A(h; \rho , q) =
1
PQ
v1\int
u1
v1\int
u1
\rho (x1)q(x2)h(x1, x2)dx1dx2 (15)
and
\| \nabla h; \rho , q\| 2 =
\left( v1\int
u1
v2\int
u2
\Biggl[
P
Q
q(x2)
\rho (x1)
\biggl(
\partial h
\partial x1
\biggr) 2
+
Q
P
\rho (x1)
\rho (x2)
\biggl(
\partial h
\partial x2
\biggr) 2
\Biggr]
dx1dx2
\right) 1
2
. (16)
Theorem 5.3. Let \varphi : I \subset \BbbR \rightarrow \BbbR be differentiable mapping with continuous first derivative.
Let h : (u1, v1)\times (u2, v2) \rightarrow I be as defined in Theorem 5.2 such that h, \varphi \circ h, \varphi \prime \circ h \in L((u1, v1)\times
\times (u2, v2)), and suppose that there exist \psi , \Psi \in \BbbR such that
\psi \leq \varphi \prime (\zeta ) \leq \Psi for all \zeta \in I.
Then, for
\rho \in C(u1, v1), q \in C(u2, v2), \rho > 0, q > 0,
v1\int
u1
\rho (t)dt = P <\infty ,
v2\int
u2
q(t)dt = Q <\infty ,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1664 S. I. BUTT, D. PEČARIĆ, J. PEČARIĆ
we have the following refinements:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1
PQ
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)(\varphi \circ h)(\zeta 1, \zeta 2)d\zeta 1d\zeta 2 - \varphi
\left( 1
PQ
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)h(\zeta 1, \zeta 2)d\zeta 1d\zeta 2
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \Psi - \psi
2
\Biggl[
1
PQ
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)h
2(\zeta 1, \zeta 2)d\zeta 1d\zeta 2 -
-
\Biggl(
1
PQ
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)h(\zeta 1, \zeta 2)d\zeta 1d\zeta 2
\Biggr) 2\Biggr] 1
2
\leq
\leq \Psi - \psi
2\pi
\| \nabla h; \rho , q\| 2,
where \| \nabla h; \rho , q\| 2 is given in (16).
Proof. Employing the mean-value theorem for points c, d \in I, we can write that there exists \xi ,
c \leq \xi \leq d, such that
\varphi (c) - \varphi (d) = \varphi \prime (\xi )(c - d). (17)
Using (17) for
c = \=h =
1
PQ
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2) h(\zeta 1, \zeta 2)d\zeta 1d\zeta 2
and d = h(\zeta 1, \zeta 2), we conclude that there exists g, \=h \leq g \leq h, such that
\varphi
\bigl(
\=h
\bigr)
- \varphi (h) = \varphi \prime (g)
\bigl(
\=h - h
\bigr)
. (18)
Now multiplying (18) by \rho (\zeta 1) and q(\zeta 2) and integrating over (u1, v1) and (u2, v2) yields
PQ\varphi
\left( 1
PQ
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)h(\zeta 1, \zeta 2)d\zeta 1d\zeta 2
\right) -
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)\varphi (h(\zeta 1, \zeta 2))d\zeta 1d\zeta 2 =
= \=h
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)\varphi
\prime (g(\zeta 1, \zeta 2))d\zeta 1d\zeta 2 -
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)\varphi
\prime (g(\zeta 1, \zeta 2))h(\zeta 1, \zeta 2)d\zeta 1d\zeta 2.
Dividing by PQ, we get
1
PQ
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)\varphi (h(\zeta 1, \zeta 2))d\zeta 1d\zeta 2 -
- \varphi
\left( 1
PQ
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)h(\zeta 1, \zeta 2)d\zeta 1d\zeta 2
\right) =
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SEVERAL JENSEN – GRÜSS INEQUALITIES WITH APPLICATIONS IN INFORMATION THEORY 1665
=
1
PQ
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)\varphi
\prime (g(\zeta 1, \zeta 2))h(\zeta 1, \zeta 2)d\zeta 1d\zeta 2 -
- 1
PQ
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)h(\zeta 1, \zeta 2)d\zeta 1d\zeta 2
1
PQ
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)\varphi
\prime (g(\zeta 1, \zeta 2))d\zeta 1d\zeta 2.
Now taking modulus on both sides and using representation \frakC (h, g; \rho , q) given in (14), we obtain\bigm| \bigm| \bigm| \bigm| \bigm| 1
PQ
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)\varphi (h(\zeta 1, \zeta 2))d\zeta 1d\zeta 2 -
- \varphi
\left( 1
PQ
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)h(\zeta 1, \zeta 2)d\zeta 1d\zeta 2
\right) \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| 1
PQ
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)\varphi
\prime (g(\zeta 1, \zeta 2))h(\zeta 1, \zeta 2)d\zeta 1d\zeta 2 -
- 1
PQ
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)h(\zeta 1, \zeta 2)d\zeta 1d\zeta 2
1
PQ
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)\varphi
\prime (g(\zeta 1, \zeta 2))d\zeta 1d\zeta 2
\bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \frakC \bigl( h, \varphi \prime (g); \rho , q
\bigr) \bigm| \bigm| .
Now applying Cauchy – Schwartz inequality, we can state that the last expression is less than\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1
PQ
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)\varphi (h(\zeta 1, \zeta 2))d\zeta 1d\zeta 2 - \varphi
\left( 1
PQ
v1\int
u1
v2\int
u2
\rho (\zeta 1)q(\zeta 2)h(\zeta 1, \zeta 2)d\zeta 1d\zeta 2
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \frakC
1
2 (h, h; \rho , q)\frakC
1
2
\bigl(
\varphi \prime (g), \varphi \prime (g); \rho , q
\bigr)
\leq \Psi - \psi
2
\frakC
1
2 (h, h; \rho , q). (19)
Finally, employing Theorem 5.2 on the second expression of (19), we complete the proof of Theo-
rem 5.3.
Remark 5.1. It is of worth mentioning that using result given in Remark 9 in [20], we can also
give extension for function h with more variables.
6. Jensen – Grüss inequality and monotonic functions. S. Bernstein in [5] introduce the term
absolutely monotonic function on interval [u, v], if h \in Ck[u, v] and satisfies
h(k)(\zeta ) \geq 0, k = 0, 1, . . . , \zeta \in (u, v),
and completely monotonic function if
( - 1)kh(k)(\zeta ) \geq 0, k = 0, 1, . . . , \zeta \in (u, v).
G. Grüss in [8] also gave results for monotone functions given as:
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1666 S. I. BUTT, D. PEČARIĆ, J. PEČARIĆ
If \varphi and h are absolutely monotone functions on (0, 1) satisfying (3), then
\frakC (\varphi ,\varphi ) \leq 4
45
(\Lambda - \lambda )2 (20)
and
| \frakC (\varphi , h)| \leq 4
45
(\Lambda - \lambda )(\Psi - \psi ). (21)
The constant 4/45 is best possible both for (20) and (21) and this can been seen by putting \varphi (u) =
= u2. G. Grüss used Bernstein’s polynomial to prove (20) and (21). However, E. Landau [14] gave
an easy proof by using his proposition (see [12, p. 297]). In [15], he also proved that inequalities
(20) and (21) still hold provided that the functions \varphi and h are monotonic of order 4. He also proved
bounds for Chebyshev functional for monotone functions of order k = 1, 2, 3 respectively as:
| \frakC (\varphi , h)| \leq 1
4
(\Lambda - \lambda )(\Psi - \psi ) for k = 1, (22)
| \frakC (\varphi , h)| \leq 1
9
(\Lambda - \lambda )(\Psi - \psi ) for k = 2, (23)
and
| \frakC (\varphi , h)| \leq 9
100
(\Lambda - \lambda )(\Psi - \psi ) for k = 3. (24)
G. Hardy [16] obtained the following result:
Let \varphi and h be totally monotonic function on (0,\infty ) and \varphi , h \in L(0, v) satisfying (3). Then
the following inequality is valid:
| \frakC (\varphi , h)| \leq 1
12
(\Lambda - \lambda )(\Psi - \psi ). (25)
Now we give several Jensen – Grüss inequalities for monotone functions.
Theorem 6.1. Let \varphi : I = (0, 1) \subset \BbbR \rightarrow \BbbR be differentiable mapping with continuous first
derivative. Let h : (0, 1) \rightarrow I be absolutely monotone function on (0, 1) such that h, \varphi \circ h, \varphi \prime \circ h \in
\in L(0, 1), and suppose that there exist \lambda , \Lambda , \psi , \Psi \in \BbbR such that
\lambda \leq h(\zeta ) \leq \Lambda , \psi \leq \varphi \prime (\zeta ) \leq \Psi for all \zeta \in I.
Then we have the following refinements:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1\int
0
h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \Psi - \psi
2
\left( 1\int
0
h2(\zeta )d\zeta -
\left( 1\int
0
h(\zeta )d\zeta
\right) 2
\right)
1
2
\leq (\Lambda - \lambda )(\Psi - \psi )
3
\surd
5
.
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SEVERAL JENSEN – GRÜSS INEQUALITIES WITH APPLICATIONS IN INFORMATION THEORY 1667
Proof. Now from the proof of Theorem 2.1 without weights, putting (u, v) = (0, 1) and \rho (\zeta ) = 1
for all \zeta \in (0, 1), we get \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1\int
0
h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
\varphi \prime (g(\zeta ))h(\zeta )d\zeta -
1\int
0
h(\zeta )d\zeta
1\int
0
\varphi \prime (g(\zeta ))d\zeta
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \frakC (h, \varphi \prime (g))
\bigm| \bigm| \leq 1
2
1\int
0
1\int
0
(\| h(\zeta ) - h(\tau )| )
\bigl( \bigm| \bigm| \varphi \prime (g(\zeta )) - \varphi \prime (g(\tau ))
\bigm| \bigm| \bigr) d\zeta d\tau .
Now applying Cauchy – Buniakowsky – Schwartz inequality for double integrals, we can state that
the last expression is less than\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1\int
0
h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq \frakC
1
2 (h, h)\frakC
1
2
\bigl(
\varphi \prime (g), \varphi \prime (g)
\bigr)
.
Now utilizing Grüss inequality (2) without weights on second term, we get
\leq \frakC
1
2 (h, h)
1
2
(\Psi - \psi ) =
\left( 1\int
0
h2(\zeta )d\zeta -
\left( 1\int
0
h(\zeta )d\zeta
\right) 2
\right)
1
2
\Psi - \psi
2
.
Now utilizing Grüss type inequality (20) for h to be absolutely monotone functions on (0, 1) on first
term, we have
\leq
\biggl(
2
3
\surd
5
(\Lambda - \lambda )
\biggr)
\Psi - \psi
2
.
Theorem 6.1 is proved.
The next results entails the bounds of Jensen – Grüss type inequalities for monotonic function of
different orders.
Corollary 6.1. Under the assumptions of Theorem 6.1, let h : (0, 1) \rightarrow I be monotonic function
on (0, 1) of order k = 1, 2, 3 such that h, \varphi \circ h, \varphi \prime \circ h \in L(0, 1), and suppose that there exist \lambda ,
\Lambda , \psi , \Psi \in \BbbR such that
\lambda \leq h(\zeta ) \leq \Lambda , \psi \leq \varphi \prime (\zeta ) \leq \Psi for all \zeta \in I.
Then we have the following several refinements:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1\int
0
h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1668 S. I. BUTT, D. PEČARIĆ, J. PEČARIĆ
\leq \Psi - \psi
2
\left( 1\int
0
h2(\zeta )d\zeta -
\left( 1\int
0
h(\zeta )d\zeta
\right) 2
\right)
1
2
\leq
\leq (\Lambda - \lambda )(\Psi - \psi )
4
for k = 1,\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1\int
0
h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \Psi - \psi
2
\left( 1\int
0
h2(\zeta )d\zeta -
\left( 1\int
0
h(\zeta )d\zeta
\right) 2
\right)
1
2
\leq
\leq (\Lambda - \lambda )(\Psi - \psi )
6
for k = 2,
and \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1\int
0
h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \Psi - \psi
2
\left( 1\int
0
h2(\zeta )d\zeta -
\left( 1\int
0
h(\zeta )d\zeta
\right) 2
\right)
1
2
\leq
\leq 3
20
(\Lambda - \lambda )(\Psi - \psi ) for k = 3. (26)
Proof. We establish the proof, when h is monotonic of order k = 3. We have already established
in the proof of Theorem 6.1 that\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1\int
0
(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1\int
0
h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq \frakC
1
2 (h, h)\frakC
1
2
\bigl(
\varphi \prime (g), \varphi \prime (g)
\bigr)
\leq
\leq \frakC
1
2 (h, h)
1
2
(\Psi - \psi ) =
\left( 1\int
0
h2(\zeta )d\zeta -
\left( 1\int
0
h(\zeta )d\zeta
\right) 2
\right)
1
2
\Psi - \psi
2
.
Now utilizing Grüss type inequality (24) for h = \varphi on first term, we get
\leq
\biggl(
3
10
(\Lambda - \lambda )
\biggr)
\Psi - \psi
2
and (26) is established.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
SEVERAL JENSEN – GRÜSS INEQUALITIES WITH APPLICATIONS IN INFORMATION THEORY 1669
Other cases when h is monotonic of order k = 1, 2 can be obtained analogously by applying
inequalities (22) and (23), respectively.
Corollary 6.1 is proved.
Now, we give refinements by using Grüss bounds obtained by G. Hardy for completely monotone
functions.
Theorem 6.2. Let \varphi : I = (0, v) \subset \BbbR \rightarrow \BbbR be differentiable mapping with continuous first
derivative. Let h : (0,\infty ) \rightarrow I be completely monotone function on (0,\infty ) such that h, \varphi \circ h,
\varphi \prime \circ h \in L(0, v), and suppose that there exist \lambda , \Lambda , \psi , \Psi \in \BbbR such that
\lambda \leq h(\cdot ) \leq \Lambda and \psi \leq \varphi \prime (\cdot ) \leq \Psi .
Then we have the following refinements:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
v\int
0
(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( v\int
0
h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \Psi - \psi
2
\left( v\int
0
h2(\zeta )d\zeta -
\left( v\int
0
h(\zeta )d\zeta
\right) 2\right)
1
2
\leq (\Lambda - \lambda )(\Psi - \psi )
4
\surd
3
.
Proof. From the proof of Theorem 2.1 without weights, putting (u, v) = (0, v) and \rho (\zeta ) = 1 for
all \zeta \in (0, v), we obtain \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1v
v\int
0
(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1
v
v\int
0
h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1v
v\int
0
\varphi \prime (g(\zeta ))h(\zeta )d\zeta - 1
v2
v\int
0
h(\zeta )d\zeta
v\int
0
\varphi \prime (g(\zeta ))d\zeta
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \frakC (h, \varphi \prime (g))
\bigm| \bigm| \leq 1
2v2
v\int
0
v\int
0
(\| h(\zeta ) - h(\tau )\| )
\bigl( \bigm| \bigm| \varphi \prime (g(\zeta )) - \varphi \prime (g(\tau ))
\bigm| \bigm| \bigr) d\zeta d\tau .
Now applying Cauchy – Buniakowsky – Schwartz inequality for double integrals, we can state that
the last expression is less than\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1v
v\int
0
(\varphi \circ h)(\zeta )d\zeta - \varphi
\left( 1
v
v\int
0
h(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq \frakC
1
2 (h, h)\frakC
1
2
\bigl(
\varphi \prime (g), \varphi \prime (g)
\bigr)
.
Now utilizing Grüss inequality (2) without weights on second term, we get
\leq \frakC
1
2 (h, h)
1
2
(\Psi - \psi ) =
=
\left( 1
v
v\int
0
h2(\zeta )d\zeta -
\left( 1
v
v\int
0
h(\zeta )d\zeta
\right) 2\right)
1
2
\Psi - \psi
2
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1670 S. I. BUTT, D. PEČARIĆ, J. PEČARIĆ
Now utilizing Grüss type inequality (25) for h = \varphi to be completely monotone function on first
term, we have
\leq
\biggl(
1\surd
12
(\Lambda - \lambda )
\biggr)
\Psi - \psi
2
.
Theorem 6.2 is proved.
7. Applications in information theory. Information theory is the study of data which ma-
nages the capacity, measurement and correspondence of data. The subject studies all the theoretical
problems related to information transformation over the communication channels. Being a theoretical
substance, data can’t be measured without any problem. Claude Shannon, who is these days regarded
as the “Father of Information Theory”, presented a theory in order to quantify the communication
of information [22]. Shannon’s theory deals with the problem of how to transmit information most
efficiently through a given channel. It also tackles the issues of communication security. Shannon’s
formula states that we will gain the largest amount of Shannon’s information when dealing with
systems whose individual possible outcomes are equally likely to occur. Shannon’s entropy is a
measure of the potential reduction in uncertainty in the receiver’s knowledge. Shannon’s entropy and
related measures are increasingly used in molecular ecology and population genetics, information
theory, dynamical systems and statistical physics.
Jensen’s integral inequality is importantly used to construct many information inequalities [1, 2,
9]. In this section, we present some important applications in information theory of our main results.
Consider the set of probability density functions
\scrP =
\left\{ \rho | \rho : I \rightarrow \BbbR , \rho (\zeta ) \geq 0 and
v\int
u
p(\zeta )d\zeta = 1
\right\} .
For positive probability density function \rho \in \scrP , the Shannon entropy is defined as [17]
H(\rho ) = -
v\int
u
\rho (\zeta ) \mathrm{l}\mathrm{n} \rho (\zeta )d\zeta .
Theorem 7.1. Under the assumptions of Theorem 2.1 with I = [u, v] \subset (0,\infty ), suppose that
\rho \in \scrP be positive probability distribution function, defined on I, such that there exist constants
0 < \lambda , \Lambda \leq 1 such that
\lambda \leq \rho (\zeta ) \leq \Lambda for all \zeta \in I.
Then we have the following refinements:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| H(\rho ) + \mathrm{l}\mathrm{n}
\left( v\int
u
\rho 2(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \Lambda - \lambda
2\Lambda \lambda
\left( v\int
u
\rho 3(\zeta )d\zeta -
\left( v\int
u
\rho 2(\zeta )d\zeta
\right) 2\right)
1
2
\leq (\Lambda - \lambda )2
4\Lambda \lambda
.
Proof. Substituting \varphi (\zeta ) := - \mathrm{l}\mathrm{n}(\zeta ) and h(\zeta ) = \rho (\zeta ) in (5), we will obtain our results.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
SEVERAL JENSEN – GRÜSS INEQUALITIES WITH APPLICATIONS IN INFORMATION THEORY 1671
Theorem 7.2. Under the assumptions of Theorem 3.1 with I = [u, v] \subset (0,\infty ), suppose that
\rho \in \scrP be positive probability distribution function, defined on I, such that (\rho \prime )2 \in L[u, v] and there
exist constants 0 < \lambda , \Lambda \leq 1 such that
\lambda \leq \rho (\zeta ) \leq \Lambda for all \zeta \in I.
Then we have the following refinements:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| H(\rho ) + \mathrm{l}\mathrm{n}
\left( v\int
u
\rho 2(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \Lambda - \lambda
2\Lambda \lambda
\left( v\int
u
\rho 3(\zeta )d\zeta -
\left( v\int
u
\rho 2(\zeta )d\zeta
\right) 2\right)
1
2
\leq
\leq \Lambda - \lambda
2\Lambda \lambda
\left( v\int
u
\u P (\zeta )
\bigl[
\rho \prime (\zeta )
\bigr] 2
d\zeta
\right) 1
2
.
Proof. Substituting \phi (\zeta ) := - \mathrm{l}\mathrm{n}(\zeta ) and h(\zeta ) = \rho (\zeta ) in (10), we will obtain our results.
Theorem 7.3. Under the assumptions of Theorem 4.2 with I = [u, v] \subset (0,\infty ), suppose that
\rho \in \scrP be positive probability distribution function, defined on I, such that \rho \in W 2
r (I) and there
exist constants 0 < \lambda , \Lambda \leq 1 such that
\lambda \leq \rho (\zeta ) \leq \Lambda for all \zeta \in I.
Then we have the following refinements:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| H(\rho ) + \mathrm{l}\mathrm{n}
\left( v\int
u
\rho 2(\zeta )d\zeta
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \Lambda - \lambda
2\Lambda \lambda
\left( v\int
u
\rho 3(\zeta )d\zeta -
\left( v\int
u
\rho 2(\zeta )d\zeta
\right) 2\right)
1
2
\leq
\leq \Lambda - \lambda
2\Lambda \lambda
1
\pi
\left( v\int
u
1
\rho (\zeta )
\bigl[
\rho \prime (\zeta )
\bigr] 2
d\zeta
\right) 1
2
.
Proof. Substituting \varphi (\zeta ) := - \mathrm{l}\mathrm{n}(\zeta ) and h(\zeta ) = \rho (\zeta ) in (12), we will obtain our results.
8. Conclusion. In this paper, we use Jensen’n integral difference and give bounds by using
Grüss inequality. We vary bounds by imposing conditions on the un known function h by employing
Chebyshev bounds and Chebyshev norm estimations. We also present multidimensional version of
our results. Some bounds for absolutely monotone and completely monotone functions are also
obtained. Finally, as an application we conclude our paper by giving new bounds for Shannon’s
entropy. Our results will be of general interest for many researchers.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1672 S. I. BUTT, D. PEČARIĆ, J. PEČARIĆ
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ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
|
| id | umjimathkievua-article-6554 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:28:47Z |
| publishDate | 2023 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/5f/bee1230ec82c288dc12ba0d659cda55f.pdf |
| spelling | umjimathkievua-article-65542023-01-23T14:02:43Z Several Jensen–Grüss inequalities with applications in information theory Several Jensen–Grüss inequalities with applications in information theory Butt, S. I. Pečarić, Ð. Pečarić, J. Butt, S. I. Pečarić, Ð. Pečarić, J. Jensen difference, Gr\ Primary 26D15, 26D10, 94A17 UDC 517.5 Several integral Jensen–Grüss&nbsp;&nbsp;inequalities are proved together with their refinements.&nbsp;&nbsp;Some new bounds for integral Jensen–Chebyshev&nbsp; inequality are obtained.&nbsp;The multidimensional integral variants are also presented.&nbsp;&nbsp;In addition, some integral Jensen–Grüss&nbsp;&nbsp;inequalities for monotone&nbsp; and completely monotone functions are established.&nbsp;&nbsp;Finally, as an application, we present the refinements&nbsp; for Shannon's entropy. УДК 517.5 Кілька нерівностей Йєнсена–Грюсса та їх застосування в теорії інформації Доведено кілька інтегральних нерівностей Йєнсена–Грюсса та їх уточнення.&nbsp;&nbsp;Отримано деякі нові оцінки для інтегральної нерівності&nbsp;&nbsp;Йєнсена–Чебишова.&nbsp;&nbsp;Також наведено багатовимірні інтегральні варіанти.&nbsp;&nbsp;Крім того, встановлено деякі інтегральні нерівності Йєнсена–Грюсса для монотонних і цілком монотонних функцій.&nbsp;&nbsp;Насамкінець в якості додатка наведено уточнення,&nbsp;отримані для ентропії Шеннона. Institute of Mathematics, NAS of Ukraine 2023-01-17 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6554 10.37863/umzh.v74i12.6554 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 12 (2022); 1654 - 1672 Український математичний журнал; Том 74 № 12 (2022); 1654 - 1672 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6554/9342 Copyright (c) 2023 SAAD BUTT, Dilda Pecaric, Josip Pecaric |
| spellingShingle | Butt, S. I. Pečarić, Ð. Pečarić, J. Butt, S. I. Pečarić, Ð. Pečarić, J. Several Jensen–Grüss inequalities with applications in information theory |
| title | Several Jensen–Grüss inequalities with applications in information theory |
| title_alt | Several Jensen–Grüss inequalities with applications in information theory |
| title_full | Several Jensen–Grüss inequalities with applications in information theory |
| title_fullStr | Several Jensen–Grüss inequalities with applications in information theory |
| title_full_unstemmed | Several Jensen–Grüss inequalities with applications in information theory |
| title_short | Several Jensen–Grüss inequalities with applications in information theory |
| title_sort | several jensen–grüss inequalities with applications in information theory |
| topic_facet | Jensen difference Gr\ Primary 26D15 26D10 94A17 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6554 |
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