Jensen – Ostrowski inequalities and integration schemes via the Darboux expansion
By using the Darboux formula obtained as a generalization of the Taylor formula, we deduce some Jensen – Ostrowski-type inequalities. The applications to quadrature rules and $f$ -divergence measures (specifically, for higher-order $\chi$ -divergence) are also given.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507619722002432 |
|---|---|
| author | Cerone, P. Dragomir, S. S. Kikianty, E. Цероне, П. Драгомир, С. С. Кікіанті, Е. |
| author_facet | Cerone, P. Dragomir, S. S. Kikianty, E. Цероне, П. Драгомир, С. С. Кікіанті, Е. |
| author_sort | Cerone, P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:25:58Z |
| description | By using the Darboux formula obtained as a generalization of the Taylor formula, we deduce some Jensen – Ostrowski-type
inequalities. The applications to quadrature rules and $f$ -divergence measures (specifically, for higher-order $\chi$ -divergence)
are also given. |
| first_indexed | 2026-03-24T02:12:12Z |
| format | Article |
| fulltext |
UDC 517.5
P. Cerone (La Trobe Univ., Australia),
S. S. Dragomir (School Eng. and Sci., Victoria Univ., Australia; School Comput. Sci. and Appl. Math., Univ.
Witwatersrand, Johannesburg, South Africa),
E. Kikianty (Dep. Math. and Appl. Math., Univ. Pretoria, South Africa)
JENSEN – OSTROWSKI INEQUALITIES AND INTEGRATION SCHEMES
VIA THE DARBOUX EXPANSION
НЕРIВНОСТI ДЖЕНСЕНА – ОСТРОВСЬКОГО ТА СХЕМИ IНТЕГРУВАННЯ
ЧЕРЕЗ РОЗКЛАД ДАРБУ
By using the Darboux formula obtained as a generalization of the Taylor formula, we deduce some Jensen – Ostrowski-type
inequalities. The applications to quadrature rules and f -divergence measures (specifically, for higher-order \chi -divergence)
are also given.
За допомогою формули Дарбу, що є узагальненням формули Тейлора, виведено деякi нерiвностi типу Дженсе-
на – Островського. Наведено також застосування до квадратурних правил та f -дивергентних мiр (зокрема, для
\chi -дивергенцiї високого порядку).
1. Introduction. In 1938, Ostrowski proved the following inequality [14]: Let f : [a, b] \rightarrow \BbbR be
continuous on [a, b] and differentiable on (a, b) such that f \prime : (a, b) \rightarrow \BbbR is bounded on (a, b), i.e.,
\| f \prime \| \infty := \mathrm{s}\mathrm{u}\mathrm{p}t\in (a,b) | f \prime (t)| < \infty . Then\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f(x) - 1
b - a
b\int
a
f(t) dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\Biggl[
1
4
+
\biggl(
x - (a+ b)/2
b - a
\biggr) 2\Biggr] \bigm\| \bigm\| f \prime \bigm\| \bigm\|
\infty (b - a)
for all x \in [a, b] and the constant 1/4 is the best possible. In particular, when x = (a + b)/2, this
inequality gives an error estimate to the midpoint rule:
\int b
a
f(t) dt \approx (b - a)f((a+ b)/2).
The midpoint rule is the simplest form of quadrature rules. Derivative-based quadrature rules
are of interest due to the larger number of parameters which increases the precision and order of
accuracy (cf. Burg [2]). Wiersma [18] introduced a derivative-based quadrature rule that is similar to
the Euler – Maclaurin formula.
In Wang and Guo [17], the Euler – Maclaurin formula, or simply Euler’s formula, is derived from
Darboux’s formula.
Proposition 1 (Darboux’s formula). Let f(z) be an analytic function along the straight line from
a point a to the point z, and \varphi (t) be an arbitrary polynomial of degree n. Then
\varphi (n)(0) [f(z) - f(a)] =
=
n\sum
m=1
( - 1)m - 1(z - a)m
\Bigl[
\varphi (n - m)(1)f (m)(z) - \varphi (n - m)(0)f (m)(a)
\Bigr]
+
+( - 1)n(z - a)n+1
1\int
0
\varphi (t)f (n+1)
\bigl[
(1 - t)a+ tz
\bigr]
dt. (1)
c\bigcirc P. CERONE, S. S. DRAGOMIR, E. KIKIANTY, 2017
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8 1123
1124 P. CERONE, S. S. DRAGOMIR, E. KIKIANTY
Taylor’s formula is a special case with \varphi (t) = (t - 1)n [17].
In [5], some inequalities are derived by utilising Taylor’s formula (with integral remainder)
f (x) = f(a) +
n\sum
k=1
(x - a)k
k!
f (k)(a) +
1
n!
x\int
a
(x - t)nf (n+1)(t) dt.
These inequalities both generalise Ostrowski’s and Jensen’s inequalities for general integrals (and
are referred to as Jensen – Ostrowski-type inequalities). In particular, an Ostrowski-type inequality in
[5, p. 68] gives the following quadrature rule:
b\int
a
f(t) dt \approx (b - a)f(\zeta ) +
n\sum
k=1
f (k)(\zeta )
(b - \zeta )k+1 - (a - \zeta )k+1
(k + 1)!
for \zeta \in [a, b] and the error estimate is given by
\| f (n+1)\| [a,b],\infty
(\zeta - a)n+2 + (b - \zeta )n+2
(n+ 2)!
.
For further reading on this type of inequalities, we refer the readers to [3 – 5, 8 – 10].
In this paper, we provide further, wider, and fuller treatment of our earlier work in [5] by
considering Darboux’s formula in place of Taylor’s formula. The work also develops broader and
more general application in areas such as derivative-based quadrature rules and divergence measures
(specifically for the higher-order \chi -divergence) as demonstrated in Sections 4 and 5, respectively.
2. Preliminaries. 2.1. Euler’s formula. This subsection serves as a reference point for the
facts concerning Euler’s formula. The explicit expression for the Bernoulli polynomial is
\varphi n(x) =
n\sum
k=0
\biggl(
n
k
\biggr)
\varphi kx
n - k,
where \varphi 0 = 1, and
n - 1\sum
k=0
1
k!(n - k)!
\varphi k = 0, n \geq 2.
The Bernoulli numbers are given by
\varphi 0 = 1, \varphi 1 = - 1
2
, \varphi 2k = ( - 1)k - 1Bk, and \varphi 2k+1 = 0, k \geq 2.
The first five Bernoulli numbers and polynomials are given in the following:
B1 =
1
6
, B2 =
1
30
, B3 =
1
42
, B4 =
1
30
, B5 =
5
66
,
\varphi 0(x) = 1, \varphi 1(x) = x - 1
2
, \varphi 2(x) = x2 - x+
1
6
,
\varphi 3(x) = x3 - 3
2
x2 +
1
2
x, \varphi 4(x) = x4 - 2x3 + x2 - 1
30
.
Choosing the Bernoulli polynomial \varphi n(t) in place of \varphi (t) and replacing n with 2n and the
polynomial \varphi n with \varphi 2n in Darboux’s formula (1) gives Euler’s formula
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
JENSEN – OSTROWSKI INEQUALITIES AND INTEGRATION SCHEMES VIA THE DARBOUX EXPANSION 1125
f(z) - f(a) =
z - a
2
\bigl[
f \prime (z) + f \prime (a)
\bigr]
+
n\sum
k=1
( - 1)k
(z - a)2k
(2k)!
Bk
\bigl[
f (2k)(z) - f (2k)(a)
\bigr]
+
+
(z - a)2n+1
(2n)!
1\int
0
\varphi 2n(t)f
(2n+1)((1 - t)a+ tz) dt. (2)
2.2. Identities. Throughout the paper, let (\Omega ,\scrA , \mu ) be a measurable space with
\int
\Omega
d\mu = 1,
consisting of a set \Omega , a \sigma -algebra \scrA of subsets of \Omega , and a countably additive and positive measure
\mu on \scrA with values in the set of extended real numbers. Throughout this subsection, let I be an
interval in \BbbR .
Lemma 1. Let f : I \rightarrow \BbbC be such that f (n) is absolutely continuous on I and a \in \r I. Let \varphi (t)
be an arbitrary polynomial of degree exactly n. If g : \Omega \rightarrow I is Lebesgue \mu -measurable on \Omega , f \circ g,
(g - a)m, (g - a)m(f (m) \circ g) \in L(\Omega , \mu ) for all m \in \{ 1, . . . , n+ 1\} , then we have\int
\Omega
f \circ g d\mu - f(a) = Pn,\varphi (a, \lambda ) +Rn,\varphi (a, \lambda ) (3)
for all \lambda \in \BbbC , where Pn,\varphi (a, \lambda ) = Pn,\varphi (a, \lambda ; f, g) is defined by
Pn,\varphi (a, \lambda ) =
1
\varphi (n)(0)
n\sum
m=1
( - 1)m - 1\times
\times
\left\{ \varphi (n - m)(1)
\int
\Omega
(g - a)m(f (m) \circ g) d\mu - \varphi (n - m)(0)f (m)(a)
\int
\Omega
(g - a)m d\mu
\right\} +
+
( - 1)n\lambda
\varphi (n)(0)
1\int
0
\varphi (t) dt
\int
\Omega
(g - a)n+1 d\mu (4)
and Rn,\varphi (a, \lambda ) = Rn,\varphi (a, \lambda ; f, g) is defined by
Rn,\varphi (a, \lambda )) =
( - 1)n
\varphi (n)(0)
\int
\Omega
(g - a)n+1
\left( 1\int
0
\varphi (t)
\Bigl[
f (n+1)[(1 - t)a+ tg] - \lambda
\Bigr]
dt
\right) d\mu =
=
( - 1)n
\varphi (n)(0)
1\int
0
\varphi (t)
\int
\Omega
(g - a)n+1
\Bigl( \Bigl[
f (n+1)[(1 - t)a+ tg] - \lambda
\Bigr]
d\mu
\Bigr)
dt. (5)
Proof. Since f (n) is absolutely continuous on I, f (n+1) exists almost everywhere on I and is
Lebesgue integrable on I. By Proposition 1, we have
f(z) - f(a) =
1
\varphi (n)(0)
n\sum
m=1
( - 1)m - 1(z - a)m\{ \varphi (n - m)(1)f (m)(z) -
- \varphi (n - m)(0)f (m)(a)\} + \lambda ( - 1)n(z - a)n+1
\varphi (n)(0)
1\int
0
\varphi (t) dt+
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
1126 P. CERONE, S. S. DRAGOMIR, E. KIKIANTY
+
( - 1)n(z - a)n+1
\varphi (n)(0)
1\int
0
\varphi (t)
\Bigl[
f (n+1)[(1 - t)a+ tz] - \lambda
\Bigr]
dt.
By replacing z with g(t) and integrating on \Omega , we obtain\int
\Omega
f \circ g d\mu - f(a) =
1
\varphi (n)(0)
n\sum
m=1
( - 1)m - 1\times
\times
\left\{ \varphi (n - m)(1)
\int
\Omega
(g - a)m(f (m) \circ g) d\mu - \varphi (n - m)(0)f (m)(a)
\int
\Omega
(g - a)m d\mu
\right\} +
+
( - 1)n\lambda
\varphi (n)(0)
1\int
0
\varphi (t) dt
\int
\Omega
(g - a)n+1 d\mu +
+
( - 1)n
\varphi (n)(0)
\int
\Omega
(g - a)n+1
\left( 1\int
0
\varphi (t)
\bigl[
f (n+1)[(1 - t)a+ tg] - \lambda
\bigr]
dt
\right) d\mu .
The last equality in (5) follows by Fubini’s theorem.
Lemma 2. Let f : I \rightarrow \BbbC be such that f (2n) is absolutely continuous on I and a \in \r I. Let
\varphi 2n(t) be the Bernoulli polynomials. If g : \Omega \rightarrow I is Lebesgue \mu -measurable on \Omega , f \circ g, (g - a)m,
(g - a)m(f (m) \circ g) \in L(\Omega , \mu ) for all m \in \{ 1, . . . , 2n+ 1\} , then we have\int
\Omega
f \circ g d\mu - f(a) = Pn(a, \lambda ) +Rn(a, \lambda )
for all \lambda \in \BbbC , where Pn(a, \lambda ) = Pn(a, \lambda ; f, g) is defined by
Pn(a, \lambda ) =
\int
\Omega
g - a
2
\bigl[
f \prime (a) + f \prime \circ g
\bigr]
d\mu +
+
\int
\Omega
n\sum
k=1
( - 1)kBk(g - a)2k
(2k)!
\bigl[
f (2k) \circ g - f (2k)(a)
\bigr]
d\mu +
+\lambda
1\int
0
\varphi 2n(t) dt
\int
\Omega
(g - a)2n+1
(2n)!
d\mu (6)
and Rn(a, \lambda ) = Rn(a, \lambda ; f, g) is defined by
Rn(a, \lambda ) =
\int
\Omega
(g - a)2n+1
(2n)!
\left[ 1\int
0
\varphi 2n(t)[f
(2n+1)((1 - t)a+ tg) - \lambda ] dt
\right] d\mu =
=
1\int
0
\varphi 2n(t)
\int
\Omega
(g - a)2n+1
(2n)!
\Bigl[
f (2n+1)((1 - t)a+ tg) - \lambda
\Bigr]
d\mu dt.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
JENSEN – OSTROWSKI INEQUALITIES AND INTEGRATION SCHEMES VIA THE DARBOUX EXPANSION 1127
The proof follows by the Euler’s formula (2) and similar arguments to those in the proof of
Lemma 1.
Remark 1. Recall that B1 = 1/6, \varphi 2(t) = t2 - t+ 1/6, and note that
\int 1
0
\varphi 2(t) dt = 0. Taking
n = 1 in Lemma 2, we get \int
\Omega
f \circ g d\mu - f(a) =
=
\int
\Omega
g - a
2
\bigl[
f \prime (a) + f \prime \circ g
\bigr]
d\mu - 1
12
\int
\Omega
(g - a)2[f \prime \prime \circ g - f \prime \prime (a)] d\mu +
+
\int
\Omega
(g - a)3
2
\left[ 1\int
0
\biggl(
t2 - t+
1
6
\biggr)
[f (3)((1 - t)a+ tg) - \lambda ] dt
\right] d\mu .
3. Main results: Jensen – Ostrowski inequalities. In this section we derive some inequalities
of Jensen – Ostrowski type using the lemmas obtain in Subsection 2.2. We use the notation
\| k\| \Omega ,p :=
\left\{
\left( \int
\Omega
| k(t)| p d\mu (t)
\right) 1/p, p \geq 1, k \in Lp(\Omega , \mu ),
\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}
t\in \Omega
| k(t)| , p = \infty , k \in L\infty (\Omega , \mu ),
and
\| f\| [0,1],p :=
\left\{
\left( 1\int
0
| f(s)| p ds
\right) 1/p, p \geq 1 f \in Lp([0, 1]),
\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}
s\in [0,1]
| f(s)| , p = \infty , f \in L\infty ([0, 1]).
We also denote by \ell , the identity function on [0, 1], namely \ell (t) = t, for t \in [0, 1].
Throughout this section, let I be an interval in \BbbR . We note that I is not necessarily a finite
interval and therefore we make the following assumptions for functions f and g for a fixed n \in \BbbN :
(A1) Let f : I \rightarrow \BbbC be such that f (n) is locally absolutely continuous on I, i.e., it is locally
absolutely continuous on each closed subinterval [a, b] on I, and a \in \r I.
(A2) Let g : \Omega \rightarrow I be Lebesgue \mu -measurable on \Omega and f \circ g, (g - a)m, (g - a)m(f (m) \circ g) \in
\in L(\Omega , \mu ) for all m \in \{ 1, . . . , n+ 1\} .
(A3) We assume that
\bigm\| \bigm\| f (n+1)[(1 - \ell )a+ \ell g] - \lambda
\bigm\| \bigm\|
[0,1],\infty < \infty for all t \in \Omega and \lambda \in \BbbC .
Furthermore, the following cases are considered for a given n \in \BbbN :
(C1)
\bigm\| \bigm\| | g - a| n+1
\bigm\| \bigm\|
\Omega ,\infty < \infty and
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| f (n+1)[(1 - \ell )a+ \ell g] - \lambda
\bigm\| \bigm\|
[0,1],\infty
\bigm\| \bigm\| \bigm\|
\Omega ,1
< \infty ;
(C2)
\bigm\| \bigm\| | g - a| n+1
\bigm\| \bigm\|
\Omega ,p
< \infty and
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| f (n+1)[(1 - \ell )a+ \ell g] - \lambda
\bigm\| \bigm\|
[0,1],\infty
\bigm\| \bigm\| \bigm\|
\Omega ,q
< \infty , where p > 1
with 1/p+ 1/q = 1;
(C3)
\bigm\| \bigm\| | g - a| n+1
\bigm\| \bigm\|
\Omega ,1
< \infty and
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| f (n+1)[(1 - \ell )a+ \ell g] - \lambda
\bigm\| \bigm\|
[0,1],\infty
\bigm\| \bigm\| \bigm\|
\Omega ,\infty
< \infty .
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
1128 P. CERONE, S. S. DRAGOMIR, E. KIKIANTY
Theorem 1. Let f and g be functions that satisfy (A1) – (A3), and \varphi (t) be an arbitrary polyno-
mial of degree n. Then
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\Omega
f \circ g d\mu - f(a) - Pn,\varphi (a, \lambda )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
1\int
0
| \varphi (t)|
| \varphi (n)(0)|
dt
\left( \int
\Omega
| g - a| n+1\| fn+1,g(a, \lambda )\| [0,1],\infty d\mu
\right) \leq
\leq
\left( 1\int
0
| \varphi (t)|
| \varphi (n)(0)|
dt
\right)
\left\{
\bigm\| \bigm\| | g - a| n+1
\bigm\| \bigm\|
\Omega ,\infty
\bigm\| \bigm\| \bigm\| \| fn+1,g(a, \lambda )\| [0,1],\infty
\bigm\| \bigm\| \bigm\|
\Omega ,1
, if (C1) holds;\bigm\| \bigm\| | g - a| n+1
\bigm\| \bigm\|
\Omega ,p
\bigm\| \bigm\| \bigm\| \| fn+1,g(a, \lambda )\| [0,1],\infty
\bigm\| \bigm\| \bigm\|
\Omega ,q
, if (C2) holds;\bigm\| \bigm\| | g - a| n+1
\bigm\| \bigm\|
\Omega ,1
\bigm\| \bigm\| \bigm\| \| fn+1,g(a, \lambda )\| [0,1],\infty
\bigm\| \bigm\| \bigm\|
\Omega ,\infty
, if (C3) holds,
for any \lambda \in \BbbC , where fn+1,g(a, \lambda ) = f (n+1)[(1 - \ell )a+ \ell g] - \lambda . Here Pn,\varphi (a, \lambda ) is as defined in (4).
Proof. Taking the modulus in (3) for any \lambda \in \BbbC , we have
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\Omega
f \circ g d\mu - f(a) - Pn,\varphi (a, \lambda )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
1\int
0
| \varphi (t)|
| \varphi (n)(0)|
\left( \int
\Omega
| g - a| n+1
\bigm| \bigm| \bigm| f (n+1)[(1 - t)a+ tg] - \lambda
\bigm| \bigm| \bigm| d\mu
\right) dt \leq
\leq
1\int
0
| \varphi (t)|
| \varphi (n)(0)|
dt
\left( \int
\Omega
| g - a| n+1
\bigm\| \bigm\| \bigm\| f (n+1)[(1 - \ell )a+ \ell g] - \lambda
\bigm\| \bigm\| \bigm\|
[0,1],\infty
d\mu
\right) .
We obtain the desired result by applying Hölder inequality.
Corollary 1. Under the assumptions of Theorem 1, if \| f (n+1)\| I,\infty < \infty , then
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\Omega
f \circ g d\mu - f(a) - Pn,\varphi (a, 0)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\left( 1\int
0
| \varphi (t)|
| \varphi (n)(0)|
dt
\right) \| f (n+1)\| I,\infty
\left( \int
\Omega
| g - a| n+1 d\mu
\right) .
Here Pn,\varphi (a, \lambda ) is as defined in (4).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
JENSEN – OSTROWSKI INEQUALITIES AND INTEGRATION SCHEMES VIA THE DARBOUX EXPANSION 1129
Proof. Let \lambda = 0 in (3), and take the modulus to obtain\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\Omega
f \circ g d\mu - f(a) - Pn,\varphi (a, 0)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
1\int
0
| \varphi (t)|
| \varphi (n)(0)|
dt
\left( \int
\Omega
| g - a| n+1
\bigm\| \bigm\| \bigm\| f (n+1)[(1 - \ell )a+ \ell g]
\bigm\| \bigm\| \bigm\|
[0,1],\infty
d\mu
\right) . (7)
For any t \in \Omega and almost every s \in [0, 1], we have\bigm| \bigm| f (n+1) ((1 - s) a+ sg(t))
\bigm| \bigm| \leq \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}
u\in I
| f (n+1)(u)| = \| f (n+1)\| I,\infty .
Therefore, we get\bigm\| \bigm\| \bigm\| f (n+1) ((1 - \ell )a+ \ell g)
\bigm\| \bigm\| \bigm\|
[0,1],\infty
\leq \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}
s\in [0,1], t\in \Omega
\| f (n+1) ((1 - s) a+ sg(t)) \| \leq
\leq \| f (n+1)\| I,\infty . (8)
The desired inequality follows from (7) and (8).
Utilising (2) and applying similar arguments to those in Theorem 1 and Corollary 1, we have the
following results.
Theorem 2. Let f and g be functions that satisfy (A1) – (A3) for 2n instead of n, and \varphi 2n(t)
be the Bernoulli polynomials. Then\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\Omega
f \circ g d\mu - f(a) - P2n(a, \lambda )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
1\int
0
| \varphi 2n(t)|
(2n)!
dt
\int
\Omega
| g - a| 2n+1 \| f2n+1,g(a, \lambda )\| [0,1],\infty d\mu \leq
\leq
\left( 1\int
0
| \varphi 2n(t)|
(2n)!
dt
\right)
\left\{
\bigm\| \bigm\| | g - a| 2n+1
\bigm\| \bigm\|
\Omega ,\infty
\bigm\| \bigm\| \bigm\| \| f2n+1,g(a, \lambda )\| [0,1],\infty
\bigm\| \bigm\| \bigm\|
\Omega ,1
, if (C1) holds for 2n,\bigm\| \bigm\| | g - a| 2n+1
\bigm\| \bigm\|
\Omega ,p
\bigm\| \bigm\| \bigm\| \| f2n+1,g(a, \lambda )\| [0,1],\infty
\bigm\| \bigm\| \bigm\|
\Omega ,q
, if (C2) holds for 2n,\bigm\| \bigm\| | g - a| 2n+1
\bigm\| \bigm\|
\Omega ,1
\bigm\| \bigm\| \bigm\| \| f2n+1,g(a, \lambda )\| [0,1],\infty
\bigm\| \bigm\| \bigm\|
\Omega ,\infty
, if (C3) holds for 2n,
for any \lambda \in \BbbC , where f2n+1,g(a, \lambda ) = f (2n+1) ((1 - \ell )a+ \ell g) - \lambda . Here Pn(a, \lambda ) is as defined
in (6).
Corollary 2. Under the assumptions of Theorem 2, if \| f (2n+1)\| I,\infty < \infty , then\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\Omega
f \circ g d\mu - f(a) - P2n(a, 0)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\left( 1\int
0
| \varphi 2n(t)|
(2n)!
dt
\right) \| f (2n+1)\| I,\infty
\left( \int
\Omega
| g - a| 2n+1 d\mu
\right) .
Here Pn(a, \lambda ) is as defined in (6).
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1130 P. CERONE, S. S. DRAGOMIR, E. KIKIANTY
Remark 2. Setting n = 1 in Corollary 2, we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\Omega
f \circ g d\mu - f(a) -
\int
\Omega
g - a
2
\bigl[
f \prime (a) + f \prime \circ g
\bigr]
d\mu +
1
12
\int
\Omega
(g - a)2[f \prime \prime \circ g - f \prime \prime (a)] d\mu
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\| f \prime \prime \prime \| I,\infty
18
\surd
3
\int
\Omega
| g - a| 3 d\mu . (9)
The following terminology introduced in [8] will be required for alternate Jensen – Ostrowski
inequality results. For \gamma ,\Gamma \in \BbbC and [a, b] an interval of real numbers, define the sets of complex-
valued functions [8]
U[a,b](\gamma ,\Gamma ) :=
\Bigl\{
h : [a, b] \rightarrow \BbbC
\bigm| \bigm| \bigm| \mathrm{R}\mathrm{e} \Bigl[ (\Gamma - h(t))(h(t) - \gamma )
\Bigr]
\geq 0 for a.e. t \in [a, b]
\Bigr\}
and
\Delta [a,b](\gamma ,\Gamma ) :=
\biggl\{
h : [a, b] \rightarrow \BbbC
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| h(t) - \gamma + \Gamma
2
\bigm| \bigm| \bigm| \bigm| \leq 1
2
| \Gamma - \gamma | for a.e. t \in [a, b]
\biggr\}
.
We recall some results in [8] concerning the above sets.
Proposition 2. For any \gamma ,\Gamma \in \BbbC and \gamma \not = \Gamma , we have:
(i) U[a,b](\gamma ,\Gamma ) = \Delta [a,b](\gamma ,\Gamma );
(ii) U[a,b](\gamma ,\Gamma ) =
\Bigl\{
h : [a, b] \rightarrow \BbbC
\bigm| \bigm| \bigl( \mathrm{R}\mathrm{e}(\Gamma ) - \mathrm{R}\mathrm{e}(h(t))
\bigr) \bigl(
\mathrm{R}\mathrm{e}(h(t)) - \mathrm{R}\mathrm{e}(\gamma )
\bigr)
+
\bigl(
\mathrm{I}\mathrm{m}(\Gamma ) -
- \mathrm{I}\mathrm{m}(h(t))
\bigr) \bigl(
\mathrm{I}\mathrm{m}(h(t)) - \mathrm{I}\mathrm{m}(\gamma )
\bigr)
\geq 0 for a.e. t \in [a, b]
\Bigr\}
.
We refer to [8] for the proofs of these results. In a nutshell, they are consequences of the identity
1
4
| \Gamma - \gamma | 2 -
\bigm| \bigm| \bigm| \bigm| z - \gamma + \Gamma
2
\bigm| \bigm| \bigm| \bigm| 2 = \mathrm{R}\mathrm{e}
\bigl[
(\Gamma - z)(\=z - \=\gamma )
\bigr]
for all z \in \BbbC .
We have the following Jensen – Ostrowski inequality for functions with bounded higher (n+1)th
derivatives.
Theorem 3. Let f and g be functions that satisfy (A1) and (A2) and \varphi (t) be an arbitrary
polynomial of degree n. For some \gamma ,\Gamma \in \BbbC , \gamma \not = \Gamma , assume that f (n+1) \in U[a,b](\gamma ,\Gamma ) = \Delta [a,b](\gamma ,\Gamma ).
Then \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\Omega
f \circ g d\mu - f(a) - Pn,\varphi
\biggl(
a,
\gamma + \Gamma
2
\biggr) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq | \Gamma - \gamma |
2
\int
\Omega
| g - a| n+1 d\mu
1\int
0
| \varphi (t)|
| \varphi (n)(0)|
dt.
Here Pn,\varphi (a, \lambda ) is as defined in (4).
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JENSEN – OSTROWSKI INEQUALITIES AND INTEGRATION SCHEMES VIA THE DARBOUX EXPANSION 1131
Proof. Let \lambda = (\gamma + \Gamma )/2 in (3), we have\int
\Omega
f \circ g d\mu - f(a) - Pn,\varphi
\biggl(
a,
\gamma + \Gamma
2
\biggr)
=
=
( - 1)n
\varphi (n)(0)
\int
\Omega
(g - a)n+1
\left( 1\int
0
\varphi (t)
\biggl[
f (n+1)[(1 - t)a+ tg] - \gamma + \Gamma
2
\biggr]
dt
\right) d\mu .
Since f (n+1) \in \Delta [a,b](\gamma ,\Gamma ), we obtain\bigm| \bigm| \bigm| \bigm| f (n+1) ((1 - t) a+ tg) - \gamma + \Gamma
2
\bigm| \bigm| \bigm| \bigm| \leq 1
2
| \Gamma - \gamma | (10)
for almost every t \in [0, 1] and any s \in \Omega . Multiply (10) with | \varphi (t)| > 0 and integrate over [0, 1],
we get
1\int
0
| \varphi (t)|
\bigm| \bigm| \bigm| \bigm| f (n+1) ((1 - t) a+ tg) - \gamma + \Gamma
2
\bigm| \bigm| \bigm| \bigm| dt \leq 1
2
| \Gamma - \gamma |
1\int
0
| \varphi (t)| dt
for any s \in \Omega . Now, we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\Omega
f \circ g d\mu - f(a) - Pn,\varphi
\biggl(
a,
\gamma + \Gamma
2
\biggr) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\int
\Omega
| g - a| n+1
\left( 1\int
0
| \varphi (t)|
| \varphi (n)(0)|
\bigm| \bigm| \bigm| \bigm| f (n+1)[(1 - t)a+ tg] - \gamma + \Gamma
2
\bigm| \bigm| \bigm| \bigm| dt
\right) d\mu \leq
\leq | \Gamma - \gamma |
2
\int
\Omega
| g - a| n+1 d\mu
1\int
0
| \varphi (t)|
| \varphi (n)(0)|
dt.
Theorem 3 is proved.
Similarly, we have the following via Euler’s formula (2) and Lemma 2. We omit the proof.
Theorem 4. Let f and g be functions that satisfy (A1) and (A2) for 2n instead of n, and \varphi 2n(t)
be the Bernoulli polynomials. For some \gamma ,\Gamma \in \BbbC , \gamma \not = \Gamma , assume that f (2n+1) \in U[a,b](\gamma ,\Gamma ) =
= \Delta [a,b](\gamma ,\Gamma ). Then \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\Omega
f \circ g d\mu - f(a) - P2n
\biggl(
a,
\gamma + \Gamma
2
\biggr) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq | \Gamma - \gamma |
2(2n)!
\int
\Omega
| g - a| 2n+1 d\mu
1\int
0
| \varphi 2n(t)| dt.
Here Pn(a, \lambda ) is as defined in (6).
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1132 P. CERONE, S. S. DRAGOMIR, E. KIKIANTY
4. Applications: quadrature rules. In this section we present quadrature rules based on the
inequalities presented in Section 3. The associated composite rules may be stated in the usual manner
by partitioning the interval [a, b] into a number of subintervals, applying the quadrature rule for each
subinterval, then adding up the results. The precise statements for these composite rules are omitted.
Let g : [a, b] \rightarrow [a, b] defined by g(t) = t and \mu (t) = t/(b - a) in Corollary 1. We have the
following quadrature rule:
b\int
a
f(t) dt \approx (b - a)f(x) +
n\sum
m=1
( - 1)m - 1\times
\times
\left\{ \varphi (n - m)(1)
\varphi (n)(0)
b\int
a
(t - x)mf (m)(t) dt - \varphi (n - m)(0)
\varphi (n)(0)
f (m)(x)
b\int
a
(t - x)m dt
\right\} =
= (b - a)f(x) +
n\sum
m=1
( - 1)m - 1\times
\times
\left\{ \varphi (n - m)(1)
\varphi (n)(0)
b\int
a
(t - x)mf (m)(t) dt - \varphi (n - m)(0)
\varphi (n)(0)
f (m)(x)
\biggl(
(b - x)m+1 - (a - x)m+1
m+ 1
\biggr) \right\}
(note that we also replace a in Corollary 1 by x) with the following error estimate:
1\int
0
| \varphi (t)|
| \varphi (n)(0)|
dt
\left( b\int
a
| t - x| n+1 dt
\right) \| f (n+1)\| [a,b],\infty =
=
1\int
0
| \varphi (t)|
| \varphi (n)(0)|
dt
\biggl(
(x - a)n+2 + (b - x)n+2
n+ 2
\biggr)
\| f (n+1)\| [a,b],\infty
for x \in [a, b].
Similarly, Corollary 2 gives us\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 32
b\int
a
f(t) dt - (b - a)f(x) - 1
2
\bigl[
(b - x)f(b) - (a - x)f(a)
\bigr]
-
- f \prime (x)
4
\bigl[
(b - x)2 - (a - x)2
\bigr]
-
-
b\int
a
n\sum
k=1
( - 1)kBk(t - x)2k
(2k)!
[f (2k)(t) - f (2k)(x)] dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\left( 1\int
0
| \varphi 2n(t)|
(2n)!
dt
\right) \| f (2n+1)\| [a,b],\infty
(x - a)2n+2 + (b - x)2n+2
2n+ 2
for all x \in [a, b], thus we have the following quadrature rule:
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JENSEN – OSTROWSKI INEQUALITIES AND INTEGRATION SCHEMES VIA THE DARBOUX EXPANSION 1133
b\int
a
f(t) dt \approx 2
3
(b - a)f(x) +
1
3
[(b - x)f(b) - (a - x)f(a)] +
+
f \prime (x)
6
\bigl[
(b - x)2 - (a - x)2
\bigr]
+
+
2
3
b\int
a
n\sum
k=1
( - 1)kBk(t - x)2k
(2k)!
\bigl[
f (2k)(t) - f (2k)(x)
\bigr]
dt
for x \in [a, b] with the following error estimate:
2
3
\left( 1\int
0
| \varphi 2n(t)|
(2n)!
dt
\right) \| f (2n+1)\| [a,b],\infty
(x - a)2n+2 + (b - x)2n+2
2n+ 2
.
When n = 1, we obtain\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 53
b\int
a
f(t) dt - (b - a)f(x) - 2
3
\bigl[
(b - x)f(b) - (a - x)f(a)
\bigr]
-
- f \prime (x)
4
\bigl[
(b - x)2 - (a - x)2
\bigr]
+
1
12
\bigl[
(b - x)2f \prime (b) - (a - x)2f \prime (a)
\bigr]
-
- f \prime \prime (x)
36
\bigl[
(b - x)3 - (a - x)3
\bigr] \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 1
72
\surd
3
\| f \prime \prime \prime \| [a,b],\infty
\bigl[
(x - a)4 + (b - x)4
\bigr]
for x \in [a, b], thus we get the following quadrature rule:
b\int
a
f(t) dt \approx 3f(x)
5
(b - a) +
2
5
[(b - x)f(b) - (a - x)f(a)] -
- 1
20
[(b - x)2f \prime (b) - (a - x)2f \prime (a)]+
+
3f \prime (x)
20
\bigl[
(b - x)2 - (a - x)2
\bigr]
+
f \prime \prime (x)
60
[(b - x)3 - (a - x)3]
for x \in [a, b] with the following error estimate:
1
120
\surd
3
\| f \prime \prime \prime \| [a,b],\infty [(x - a)4 + (b - x)4].
5. Applications for \bfitf -divergence. Assume that a set \Omega and the \sigma -finite measure \mu are given.
Consider the set of all probability densities on \mu to be
\scrP :=
\left\{ p
\bigm| \bigm| \bigm| p : \Omega \rightarrow \BbbR , p(t) \geq 0,
\int
\Omega
p(t)d\mu (t) = 1
\right\} .
We recall the definition of some divergence measures, which we use in this text.
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1134 P. CERONE, S. S. DRAGOMIR, E. KIKIANTY
Definition 1. Let p, q \in \scrP and k \geq 2.
1. The Kullback – Leibler divergence [12]:
DKL (p, q) :=
\int
\Omega
p(t) \mathrm{l}\mathrm{o}\mathrm{g}
\biggl[
p(t)
q(t)
\biggr]
d\mu (t), p, q \in \scrP .
2. The \chi 2-divergence:
D\chi 2 (p, q) :=
\int
\Omega
p(t)
\Biggl[ \biggl(
q(t)
p(t)
\biggr) 2
- 1
\Biggr]
d\mu (t), p, q \in \scrP . (11)
3. Higher order \chi -divergence [1]:
D\chi k(p, q) :=
\int
\Omega
(q(t) - p(t))k
pk - 1(t)
d\mu (t) =
\int
\Omega
\biggl(
q(t)
p(t)
- 1
\biggr) k
p(t) d\mu (t), (12)
D| \chi | k(p, q) :=
\int
\Omega
| q(t) - p(t)| k
pk - 1(t)
d\mu (t) =
\int
\Omega
\bigm| \bigm| \bigm| \bigm| q(t)p(t)
- 1
\bigm| \bigm| \bigm| \bigm| k p(t) d\mu (t). (13)
Furthermore, (12) and (13) can be generalised as follows [13]:
D\chi k,a(p, q) :=
\int
\Omega
(q(t) - ap(t))k
pk - 1(t)
d\mu (t) =
\int
\Omega
\biggl(
q(t)
p(t)
- a
\biggr) k
p(t) d\mu (t),
D| \chi | k,a(p, q) :=
\int
\Omega
| q(t) - ap(t)| k
pk - 1(t)
d\mu (t) =
\int
\Omega
\bigm| \bigm| \bigm| \bigm| q(t)p(t)
- a
\bigm| \bigm| \bigm| \bigm| k p(t) d\mu (t).
4. Csiszár f-divergence [6]:
If (p, q) :=
\int
\Omega
p(t)f
\biggl[
q(t)
p(t)
\biggr]
d\mu (t), p, q \in \scrP ,
where f is convex on (0,\infty ). It is assumed that f(u) is zero and strictly convex at u = 1.
Remark 3. (1) We note that when k = 2, (12) coincides with (11).
(2) The Kullback – Leibler divergence and the \chi 2-divergence are particular instances of Csiszár
f -divergence. For the basic properties of Csiszár f-divergence, we refer the readers to [6, 7, 16].
Example 1. (i) Let f : (0,\infty ) \rightarrow \BbbR be defined by f(t) = t \mathrm{l}\mathrm{o}\mathrm{g}(t). We have
If (p, q) =
\int
\Omega
p(t)
q(t)
p(t)
\mathrm{l}\mathrm{o}\mathrm{g}
\biggl[
q(t)
p(t)
\biggr]
d\mu (t) =
\int
\Omega
q(t) \mathrm{l}\mathrm{o}\mathrm{g}
\biggl[
q(t)
p(t)
\biggr]
d\mu (t) = DKL(q, p).
(ii) Let g : (0,\infty ) \rightarrow \BbbR be defined by g(t) = - \mathrm{l}\mathrm{o}\mathrm{g}(t). We get
Ig (p, q) = -
\int
\Omega
p(t) \mathrm{l}\mathrm{o}\mathrm{g}
\biggl[
q(t)
p(t)
\biggr]
d\mu (t) =
\int
\Omega
p(t) \mathrm{l}\mathrm{o}\mathrm{g}
\biggl[
p(t)
q(t)
\biggr]
d\mu (t) = DKL(p, q).
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JENSEN – OSTROWSKI INEQUALITIES AND INTEGRATION SCHEMES VIA THE DARBOUX EXPANSION 1135
We obtain the next three results by choosing g(t) = q(t)/p(t) in Corollaries 1 and 2, and (9).
We also note that
\int
\Omega
p(t)d\mu = 1. The proofs are straightforward and therefore we omit the details.
Proposition 3. Let f : (0,\infty ) \rightarrow \BbbR be a convex function with the property that f(1) = 0.
Let \varphi (t) be an arbitrary polynomial of degree n. Assume that p, q \in \scrP and there exists constants
0 < r < 1 < R < \infty such that
r \leq q(t)
p(t)
\leq R, for \mu -a.e. t \in \Omega .
If a \in [r,R] and f (n) is absolutely continuous on [r,R], then we have the inequalities\bigm| \bigm| \bigm| \bigm| \bigm| If (p, q) - f(a) +
1
\varphi (n)(0)
n\sum
m=1
( - 1)m - 1 \times
\times
\left\{ \varphi (n - m)(0)f (m)(a)D\chi m,a(p, q) - \varphi (n - m)(1)
\int
\Omega
(q(t) - ap(t))m
pm - 1(t)
f (m)
\biggl(
q(t)
p(t)
\biggr)
d\mu
\right\}
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\left( 1\int
0
| \varphi (t)|
| \varphi (n)(0)|
dt
\right) \| f (n+1)\| [r,R],\infty D| \chi | n+1,a(p, q).
Proposition 4. Let f : (0,\infty ) \rightarrow \BbbR be a convex function with the property that f(1) = 0. Let
\varphi 2n(t) be the Bernoulli polynomials. Assume that p, q \in \scrP and there exists constants 0 < r < 1 <
< R < \infty such that
r \leq q(t)
p(t)
\leq R for \mu -a.e. t \in \Omega .
If a \in [r,R] and f (2n) is absolutely continuous on [r,R], then we have the inequalities\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| If (p, q) - f(a) - f \prime (a)
2
(1 - a) - 1
2
\int
\Omega
[q(t) - ap(t)]f \prime
\biggl(
q(t)
p(t)
\biggr)
d\mu -
-
n\sum
k=1
( - 1)kBk
(2k)!
\left[ \int
\Omega
(q(t) - ap(t))2k
p2k - 1(t)
f (2k)
\biggl(
q(t)
p(t)
\biggr)
d\mu - f (2k)(a)D\chi 2k,a(p, q)
\right] \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\left( 1\int
0
| \varphi 2n(t)|
(2n)!
dt
\right) \bigm\| \bigm\| f (2n+1)
\bigm\| \bigm\|
[r,R],\infty D| \chi | 2n+1,a(p, q).
Corollary 3. Let f : (0,\infty ) \rightarrow \BbbR be a convex function with the property that f(1) = 0. Assume
that p, q \in \scrP and there exist constants 0 < r < 1 < R < \infty such that r \leq q(t)
p(t)
\leq R, for \mu -a.e.
t \in \Omega . If a \in [r,R] and f \prime \prime is absolutely continuous on [r,R], then we have the inequalities\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| If (p, q) - f(a) - f \prime (a)
2
(1 - a) - 1
2
\int
\Omega
[q(t) - ap(t)]f \prime
\biggl(
q(t)
p(t)
\biggr)
d\mu +
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1136 P. CERONE, S. S. DRAGOMIR, E. KIKIANTY
+
1
12
\int
\Omega
(q(t) - ap(t))2
p(t)
f \prime \prime
\biggl(
q(t)
p(t)
\biggr)
d\mu - f \prime \prime (a)
12
D\chi 2,a(p, q)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 1
18
\surd
3
\| f \prime \prime \prime \| [r,R],\infty D| \chi | 3,a(p, q).
Example 2. We consider the convex function f : (0,\infty ) \rightarrow \BbbR , f(t) = t \mathrm{l}\mathrm{o}\mathrm{g}(t). We obtain
f \prime (t) = \mathrm{l}\mathrm{o}\mathrm{g}(t) + 1 and f (k)(t) = ( - 1)kt - (k - 1)(k - 2)! for k \geq 2.
Thus, \| f (k)\| [r,R] = r - (k - 1)(k - 2)!. Recall from Example 1 Part (i) that If (p, q) = DKL(q, p). We
also get \int
\Omega
(q(t) - ap(t))m
pm - 1(t)
f (m)
\biggl(
q(t)
p(t)
\biggr)
d\mu =
= ( - 1)m(m - 2)!
\int
\Omega
(q(t) - ap(t))m
pm - 1(t)
\biggl(
p(t)
q(t)
\biggr) m - 1
d\mu =
= ( - 1)m( - a)m(m - 2)!
\int
\Omega
(p(t) - 1
aq(t))
m
qm - 1(t)
d\mu =
= am(m - 2)!D\chi m, 1
a
(q, p).
Therefore, Proposition 3 gives us\bigm| \bigm| \bigm| \bigm| \bigm| DKL(q, p) - a \mathrm{l}\mathrm{o}\mathrm{g}(a) - 1
\varphi (n)(0)
n\sum
m=1
(m - 2)! \times
\times
\Biggl\{
\varphi (n - m)(0)
am - 1
D\chi m,a(p, q) + ( - 1)m - 1am\varphi (n - m)(1)D\chi m, 1
a
(q, p)
\Biggr\} \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq (n - 1)!
rn
\left( 1\int
0
| \varphi (t)|
| \varphi (n)(0)|
dt
\right) D| \chi | n+1,a(p, q).
In particular, when a = 1, we have\bigm| \bigm| \bigm| \bigm| \bigm| DKL(q, p) -
n\sum
m=1
(m - 2)!
\Biggl\{
\varphi (n - m)(0)
\varphi (n)(0)
D\chi m(p, q) +
+( - 1)m - 1\varphi
(n - m)(1)
\varphi (n)(0)
D\chi m(q, p)
\Biggr\} \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq (n - 1)!
rn
\left( 1\int
0
| \varphi (t)|
| \varphi (n)(0)|
dt
\right) D| \chi | n+1(p, q).
We also obtain
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
JENSEN – OSTROWSKI INEQUALITIES AND INTEGRATION SCHEMES VIA THE DARBOUX EXPANSION 1137\int
\Omega
q(t)f \prime
\biggl(
q(t)
p(t)
\biggr)
d\mu (t) =
\int
\Omega
\biggl(
q(t) \mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
q(t)
p(t)
\biggr)
+ q(t)
\biggr)
d\mu (t) =
=
\int
\Omega
q(t) \mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
q(t)
p(t)
\biggr)
d\mu (t) + 1 = DKL(q, p) + 1
and \int
\Omega
p(t)f \prime
\biggl(
q(t)
p(t)
\biggr)
d\mu =
\int
\Omega
p(t)
\biggl[
\mathrm{l}\mathrm{o}\mathrm{g}
\biggl[
q(t)
p(t)
\biggr]
+ 1
\biggr]
d\mu (t) =
= -
\int
\Omega
p(t) \mathrm{l}\mathrm{o}\mathrm{g}
\biggl[
p(t)
q(t)
\biggr]
d\mu (t) + 1 = - DKL(p, q) + 1.
Therefore, Proposition 4 gives us\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| DKL(q, p) - a \mathrm{l}\mathrm{o}\mathrm{g}(a) - \mathrm{l}\mathrm{o}\mathrm{g}(a) + 1
2
(1 - a) - 1
2
DKL(q, p) -
1
2
- aDKL(p, q)
2
+
+
a
2
-
n\sum
k=1
( - 1)kBk
(2k)!
(2k - 2)!
\left[ \int
\Omega
(q(t) - ap(t))2k
q2k - 1(t)
d\mu (t) -
D\chi 2k,a(p, q)
a2k - 1
\right] \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| DKL(q, p) -
1
2
\mathrm{l}\mathrm{o}\mathrm{g}(a)(a+ 1) + (a - 1) - 1
2
(DKL(q, p) + aDKL(p, q)) -
-
n\sum
k=1
( - 1)kBk
4k2 - 2k
\biggl[
a2kD\chi 2k, 1
a
(q, p) -
D\chi 2k,a(p, q)
a2k - 1
\biggr] \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq (2n - 1)!
r2n
\left( 1\int
0
| \varphi 2n(t)|
(2n)!
dt
\right) D| \chi | 2n+1,a(p, q) =
=
\left( 1\int
0
| \varphi 2n(t)|
2n
dt
\right) D| \chi | 2n+1,a(p, q)
r2n
.
In particular, when a = 1, we have\bigm| \bigm| \bigm| \bigm| \bigm| DKL(q, p) - DKL(p, q) -
n\sum
k=1
( - 1)kBk
2k2 - k
\bigl[
D\chi 2k(q, p) - D\chi 2k(p, q)
\bigr] \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\left( 1\int
0
| \varphi 2n(t)|
n
dt
\right) D| \chi | 2n+1(p, q)
r2n
. (14)
We note that \int
\Omega
(q(t) - ap(t))2
p(t)
f \prime \prime
\biggl(
q(t)
p(t)
\biggr)
d\mu (t) =
\int
\Omega
(q(t) - ap(t))2
q(t)
d\mu (t) =
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
1138 P. CERONE, S. S. DRAGOMIR, E. KIKIANTY
= 1 - 2a+ a2
\int
\Omega
p(t)2
q(t)
d\mu (t) =
= 1 - 2a+ a2(D\chi 2(q, p) + 1) =
= a2D\chi 2(q, p) + (1 - a)2.
Note the use of (14). Thus, Corollary 3 gives us\bigm| \bigm| \bigm| \bigm| DKL(q, p) -
1
2
\mathrm{l}\mathrm{o}\mathrm{g}(a)(a+ 1) + (a - 1) - 1
2
(DKL(q, p) + aDKL(p, q)) +
+
1
12
\biggl[
a2D\chi 2(q, p) + (1 - a)2 - 1
a
D\chi 2,a(p, q)
\biggr] \bigm| \bigm| \bigm| \bigm| \leq
\leq
D| \chi | 3,a(p, q)
18
\surd
3r2
.
In particular, when a = 1, we have\bigm| \bigm| \bigm| \bigm| DKL(q, p) - DKL(p, q) +
1
6
\bigl[
D\chi 2(q, p) - D\chi 2(p, q)
\bigr] \bigm| \bigm| \bigm| \bigm| \leq D| \chi | 3(p, q)
9
\surd
3r2
.
Example 3. We consider the convex function g : (0,\infty ) \rightarrow \BbbR , g(t) = - \mathrm{l}\mathrm{o}\mathrm{g}(t). We have
g(k)(t) = ( - 1)kt - k(k - 1)! for k \geq 1.
Thus, \| g(k)\| [r,R] = r - k. From Example 1 Part (ii), we have Ig(p, q) = DKL(p, q). Proposition 3
gives us \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| DKL(p, q) + \mathrm{l}\mathrm{o}\mathrm{g}(a) - 1
\varphi (n)(0)
n\sum
m=1
(m - 1)! \times
\times
\left\{ \varphi (n - m)(0)
am
D\chi m,a(p, q) - \varphi (n - m)(1)
\int
\Omega
\biggl(
1 - a
p(t)
q(t)
\biggr) m
p(t) d\mu
\right\}
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq n!
rn+1
\left( 1\int
0
| \varphi (t)|
| \varphi (n)(0)|
dt
\right) D| \chi | n+1,a(p, q).
In particular, when a = 1, we obtain\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| DKL(p, q) -
1
\varphi (n)(0)
n\sum
m=1
(m - 1)! \times
\times
\left\{ \varphi (n - m)(0)D\chi m(p, q) - \varphi (n - m)(1)
\int
\Omega
\biggl(
1 - p(t)
q(t)
\biggr) m
p(t) d\mu
\right\}
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq n!
rn+1
\left( 1\int
0
| \varphi (t)|
| \varphi (n)(0)|
dt
\right) D| \chi | n+1(p, q).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
JENSEN – OSTROWSKI INEQUALITIES AND INTEGRATION SCHEMES VIA THE DARBOUX EXPANSION 1139
We get \int
\Omega
q(t)g\prime
\biggl(
q(t)
p(t)
\biggr)
d\mu = -
\int
\Omega
q(t)
\biggl(
p(t)
q(t)
\biggr)
d\mu = - 1
and \int
\Omega
p(t)g\prime
\biggl(
q(t)
p(t)
\biggr)
d\mu = -
\int
\Omega
p2(t)
q(t)
d\mu (t) = -
\bigl[
D\chi 2(q, p) + 1
\bigr]
.
Note the use of the following identity:
D\chi 2(q, p) =
\int
\Omega
q(t)
\Biggl[ \biggl(
p(t)
q(t)
\biggr) 2
- 1
\Biggr]
d\mu (t) =
\int
\Omega
p2(t)
q(t)
d\mu (t) - 1.
Proposition 4 gives us\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| DKL(p, q) + \mathrm{l}\mathrm{o}\mathrm{g}(a) +
1
2a
(1 - a) +
1
2
- a
2
(D\chi 2(q, p) + 1) -
-
n\sum
k=1
( - 1)kBk
2k
\left[ \int
\Omega
\biggl(
1 - a
p(t)
q(t)
\biggr) 2k
p(t) d\mu - 1
a2k
D\chi 2k,a(p, q)
\right] \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\left( 1\int
0
| \varphi 2n(t)| dt
\right) D| \chi | 2n+1,a(p, q)
r2n+1
.
In particular, when a = 1, we have \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| DKL(p, q) -
1
2
D\chi 2(q, p) -
-
n\sum
k=1
( - 1)kBk
2k
\left[ \int
\Omega
\biggl(
1 - p(t)
q(t)
\biggr) 2k
p(t) d\mu - D\chi 2k(p, q)
\right] \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\left( 1\int
0
| \varphi 2n(t)| dt
\right) D| \chi | 2n+1(p, q)
r2n+1
.
Corollary 3 gives us\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| DKL(p, q) + \mathrm{l}\mathrm{o}\mathrm{g}(a) +
1
2a
(1 - a) +
1
2
- a
2
(D\chi 2(q, p) + 1) +
+
1
12
\int
\Omega
\biggl(
1 - a
p(t)
q(t)
\biggr) 2
p(t) d\mu - 1
12a2
D\chi 2,a(p, q)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
D| \chi | 3,a(p, q)
9
\surd
3r3
.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
1140 P. CERONE, S. S. DRAGOMIR, E. KIKIANTY
In particular, when a = 1, we obtain\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| DKL(p, q) -
2
3
D\chi 2(q, p) +
1
12
\left[ - 1 +
\int
\Omega
\biggl(
p(t)
q(t)
\biggr) 2
p(t) d\mu - D\chi 2(p, q)
\right] \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq D| \chi | 3(p, q)
9
\surd
3r3
.
We note the use of\int
\Omega
\biggl(
1 - p(t)
q(t)
\biggr) 2
p(t) d\mu =
\int
\Omega
\Biggl(
p(t) - 2
(p(t))2
q(t)
+
\biggl(
p(t)
q(t)
\biggr) 2
p(t)
\Biggr)
d\mu =
= 1 - 2(D\chi 2(q, p) + 1) +
\int
\Omega
\biggl(
p(t)
q(t)
\biggr) 2
p(t) d\mu =
= - 1 - 2D\chi 2(q, p) +
\int
\Omega
\biggl(
p(t)
q(t)
\biggr) 2
p(t) d\mu .
References
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Received 28.09.16,
after revision — 16.03.17
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
|
| id | umjimathkievua-article-1763 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:12:12Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c2/c44cb3193da4d5fbee1ca6a2cfdfd4c2.pdf |
| spelling | umjimathkievua-article-17632019-12-05T09:25:58Z Jensen – Ostrowski inequalities and integration schemes via the Darboux expansion Нерiвностi Дженсена–Островського та схеми iнтегрування через розклад Дарбу Cerone, P. Dragomir, S. S. Kikianty, E. Цероне, П. Драгомир, С. С. Кікіанті, Е. By using the Darboux formula obtained as a generalization of the Taylor formula, we deduce some Jensen – Ostrowski-type inequalities. The applications to quadrature rules and $f$ -divergence measures (specifically, for higher-order $\chi$ -divergence) are also given. За допомогою формули Дарбу, що є узагальненням формули Тейлора, виведено деякi нерiвностi типу Дженсена – Островського. Наведено також застосування до квадратурних правил та $f$ -дивергентних мiр (зокрема, для $\chi$ -дивергенцiї високого порядку). Institute of Mathematics, NAS of Ukraine 2017-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1763 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 8 (2017); 1123-1140 Український математичний журнал; Том 69 № 8 (2017); 1123-1140 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1763/745 Copyright (c) 2017 Cerone P.; Dragomir S. S.; Kikianty E. |
| spellingShingle | Cerone, P. Dragomir, S. S. Kikianty, E. Цероне, П. Драгомир, С. С. Кікіанті, Е. Jensen – Ostrowski inequalities and integration schemes via the Darboux expansion |
| title | Jensen – Ostrowski inequalities and integration schemes
via the Darboux expansion |
| title_alt | Нерiвностi Дженсена–Островського та схеми iнтегрування
через розклад Дарбу |
| title_full | Jensen – Ostrowski inequalities and integration schemes
via the Darboux expansion |
| title_fullStr | Jensen – Ostrowski inequalities and integration schemes
via the Darboux expansion |
| title_full_unstemmed | Jensen – Ostrowski inequalities and integration schemes
via the Darboux expansion |
| title_short | Jensen – Ostrowski inequalities and integration schemes
via the Darboux expansion |
| title_sort | jensen – ostrowski inequalities and integration schemes
via the darboux expansion |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1763 |
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