New Sharp Ostrowski-type Inequalities and Generalized Trapezoid-type Inequalities for Riemann–Stieltjes Integrals and their Applications
We prove new sharp weighted generalizations of Ostrowski-type and generalized trapezoid-type inequalities for Riemann–Stieltjes integrals. Several related inequalities are deduced and investigated. New Simpson-type inequalities are obtained for the \( \mathcal{R}\mathcal{S} \) -integral. Finally, a...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508371109543936 |
|---|---|
| author | Alomari, M. W. Аломарі, М. В. |
| author_facet | Alomari, M. W. Аломарі, М. В. |
| author_sort | Alomari, M. W. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:16:28Z |
| description | We prove new sharp weighted generalizations of Ostrowski-type and generalized trapezoid-type inequalities for Riemann–Stieltjes integrals. Several related inequalities are deduced and investigated. New Simpson-type inequalities are obtained for the \( \mathcal{R}\mathcal{S} \) -integral. Finally, as an application, we estimate the error of a general quadrature rule for the \( \mathcal{R}\mathcal{S} \) -integral via the Ostrowski–generalized-trapezoid-quadrature formula. |
| first_indexed | 2026-03-24T02:24:08Z |
| format | Article |
| fulltext |
UDC 517.5
M. W. Alomari (Jerash Univ., Jordan)
NEW SHARP INEQUALITIES OF OSTROWSKI TYPE
AND GENERALIZED TRAPEZOID TYPE FOR RIEMANN–STIELTJES
INTEGRALS AND APPLICATIONS
НОВI ТОЧНI НЕРIВНОСТI ТИПУ ОСТРОВСЬКОГО
ТА ТИПУ УЗАГАЛЬНЕНОГО ТРАПЕЦОЇДА
ДЛЯ IНТЕГРАЛIВ РIМАНА – СТIЛЬТЬЄСА ТА ЇХ ЗАСТОСУВАННЯ
We prove new sharp weighted generalizations of Ostrowski type and generalized trapezoid type inequalities for Riemann –
Stieltjes integrals. Several related inequalities are deduced and investigated. New Simpson-type inequalities for the RS-
integral obtained. Finally, as an application, an error estimate is given for a general quadrature rule for the RS-integral via
the Ostrowski — generalized trapezoid quadrature formula.
Доведено новi точнi зваженi узагальнення нерiвностей типу Островського та типу узагальненого трапецоїда для
iнтегралiв Рiмана – Стiльтьєса. Отримано та дослiджено кiлька близьких нерiвностей. Отримано новi нерiвностi
типу Сiмпсона для RS-iнтеграла. Як застосування наведено оцiнку похибки загального правила квадратур для
RS-iнтеграла iз використанням квадратурної формули Островського — узагальненого трапецоїда.
1. Introduction. In order to approximate the Riemann – Stieltjes integral
∫ b
a
f(t)du(t), Dragomir
[12] has introduced the following (general) quadrature rule:
D (f, u;x) := f(x) [u (b)− u(a)]−
b∫
a
f(t)du(t).
After that, many authors have studied this quadrature rule under various assumptions of integrands
and integrators. In the following, we give a summary of these results: let f, u : [a, b] → R be as
follow:
(1) f is of r-Hf -Hölder type on [a, b], where Hf > 0 and r ∈ (0, 1] are given,
(1′) u is of s-Hu-Hölder type on [a, b], where Hu > 0 and s ∈ (0, 1] are given,
(2) f is of bounded variation on [a, b],
(2′) u is of bounded variation on [a, b],
(3) f is Lf -Lipschitz on [a, b],
(3′) u is Lu-Lipschitz on [a, b],
(4) f is monotonic nondecreasing on [a, b],
(4′) u is monotonic nondecreasing on [a, b],
(5) f is L1,f -Lipschitz on [a, x] and L2,f -Lipschitz on [x, b],
(5′) u is L1,u-Lipschitz on [a, x] and L2,u-Lipschitz on [x, b],
(6) f is monotonic nondecreasing on [a, x] and [x, b],
(6′) u is monotonic nondecreasing on [a, x] and [x, b],
(7) f is absolutely continuous on [a, b],
(8) |f ′| is convex on [a, b].
c© M. W. ALOMARI, 2013
894 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
NEW SHARP INEQUALITIES OF OSTROWSKI TYPE AND GENERALIZED TRAPEZOID TYPE . . . 895
Then, the following inequalities hold under the corresponding assumptions:
|D (f, u;x)| ≤
≤
Hf
[
b− a
2
+
∣∣∣∣x− a+ b
2
∣∣∣∣]r∨b
a (u) , (1), (2′) [13]
Hu
[(x− a)s + (b− x)s]
[
1
2
∨b
a(f) +
1
2
∣∣∣∨x
a(f)−
∨b
x(f)
∣∣∣],
[(x− a)qs + (b− x)qs]1/q
[
(
∨x
a(f))
p +
(∨b
x(f)
)p]1/p
, (1′), (2) [14]
p > 1,
1
p
+
1
q
= 1,[
b− a
2
+
∣∣∣∣x− a+ b
2
∣∣∣∣]s∨b
a(f),
LuHf
r + 1
[
(x− a)r+1 + (b− x)r+1
]
, (1), (3′) [6]
LfHu
s+ 1
[
(x− a)s+1 + (b− x)s+1
]
, (1′), (3) [6]
LuLf
1
4
+
(
x− a+b
2
b− a
)2
(b− a)2 , (3), (3′) [6]
max {L1,u, L2,u} ×
[
b− a
2
+
∣∣∣∣x− a+ b
2
∣∣∣∣] [f(b)− f(a)] ,
(5′), (6) [6][
f(b)− f(a)
2
+
1
2
∣∣∣∣f(x)− f(a) + f(b)
2
∣∣∣∣] (b− a) ,
max {L1,f , L2,f} ×
[
b− a
2
+
∣∣∣∣x− a+ b
2
∣∣∣∣] [u(b)− u(a)] ,
(5), (6′) [6][
u(b)− u(a)
2
+
1
2
∣∣∣∣u(x)− u(a) + u(b)
2
∣∣∣∣] (b− a) ,
Hf
[
b− a
2
+
∣∣∣∣x− a+ b
2
∣∣∣∣]r [u(b)− u(a)] , (1), (4′) [11]
Hu
[
b− a
2
+
∣∣∣∣x− a+ b
2
∣∣∣∣]s [f(b)− f(a)] , (1′), (4) [11]
sup
t∈[a,x]
{(x− t)µ (f ;x, t)}
∨x
a(u) + sup
t∈[x,b]
{(t− x)µ (f ;x, t)}
∨b
x(u), (2′), (7) [7]
1
2
[
(x− a)
∨x
a(u) ‖f ′‖∞,[a,x] + (b− x)
∨b
x(u) ‖f ′‖∞,[x,b]
]
+
+
1
2
|f ′(x)|
[
(x− a)
∨x
a(u) + (b− x)
∨b
x(u)
]
. (2′), (7), (8) [7]
(1.1)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
896 M. W. ALOMARI
More details about each inequality of the above, the reader may refer to the corresponding mentioned
references and the references therein.
From a different view point, the authors of [14] considered the problem of approximating the
Stieltjes integral
∫ b
a
f(t)du(t) via the generalized trapezoid rule
T (f, u;x) := [u(x)− u(a)] f(a) + [(b)− u(x)] f(b)−
b∫
a
f(t)du(t),
|T (f, u;x)| ≤
≤
Hu
[
b− a
2
+
∣∣∣∣x− a+ b
2
∣∣∣∣]r∨b
a (f) , (1′), (2) [15]
Hf
[(x− a)s + (b− x)s]
[
1
2
∨b
a(u) +
1
2
∣∣∣∨x
a(u)−
∨b
x(u)
∣∣∣] ,
[(x− a)qs + (b− x)qs]1/q
[
(
∨x
a(u))
p +
(∨b
x(u)
)p]1/p
, (1), (2′) [8]
p > 1,
1
p
+
1
q
= 1,
[
b− a
2
+
∣∣∣∣x− a+ b
2
∣∣∣∣]s∨b
a (u) .
(1.2)
For new quadrature rules involving RS-integral see the recent works [1, 2]. For other results
concerning various approximation for RS-integral under various assumptions on f and u, see
[3, 4, 8, 9, 15 – 18] and the references therein.
In the recent work [19], Z. Liu has proved sharp generalization of weighted Ostrowski type
inequality for mappings of bounded variation, as follows (see also [20]):
Theorem 1.1. Let f : [a, b] → R be a mapping of bounded variation, g : [a, b] → [0,∞)
continuous and positive on (a, b). Then for any x ∈ [a, b] and α ∈ [0, 1], we have∣∣∣∣∣∣
b∫
a
f(t)g(t)dt−
(1− α) f(x) b∫
a
g(t)dt + α
f(a) x∫
a
g (t) dt+ f(b)
b∫
x
g(t)dt
∣∣∣∣∣∣ ≤
≤
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣]
1
2
b∫
a
g(t)dt+
∣∣∣∣∣∣
x∫
a
g(t)dt− 1
2
b∫
a
g(t)dt
∣∣∣∣∣∣
b∨
a
(f), (1.3)
where
∨b
a (f) denotes to the total variation of f over [a, b]. The constant
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] is the best
possible.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
NEW SHARP INEQUALITIES OF OSTROWSKI TYPE AND GENERALIZED TRAPEZOID TYPE . . . 897
For recent results concerning Ostrowski inequality for mappings of bounded variation see [11,
19 – 23].
The main aim in this paper, is to introduce and discuss new weighted generalizations of the
Ostrowski and the generalized trapezoid inequalities for the Riemann – Stieltjes integrals.
2. Main results. We begin with the following result:
Theorem 2.1. Let g, u : [a, b]→ [0,∞) be such that g is continuous and positive on [a, b] and
u is monotonic increasing on [a, b]. If f : [a, b] → R is a mapping of bounded variation on [a, b],
then for any x ∈ [a, b] and α ∈ [0, 1], we have∣∣∣∣∣∣∣(1− α)
f(x) (a+b)/2∫
a
g(s)du(s) + f (a+ b− x)
b∫
(a+b)/2
g(s)du(s)
+
+α
f(a) x∫
a
g(s)du(s) + f(b)
b∫
x
g(s)du(s)
− b∫
a
f(t)g(t)du(t)
∣∣∣∣∣∣ ≤
≤
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣]
1
2
b∫
a
g(t)du(t) +
∣∣∣∣∣∣
x∫
a
g(t)du(t)− 1
2
b∫
a
g(t)du(t)
∣∣∣∣∣∣
b∨
a
(f), (2.1)
where
∨b
a (f) denotes to the total variation of f over [a, b]. The constant
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] is the best
possible.
Proof. Define the mapping
Kg,u (t;x) :=
(1− α)
∫ t
a
g(s)du(s) + α
t∫
x
g(s)du(s), t ∈ [a, x] ,
(1− α)
∫ t
(a+b)/2
g(s)du(s) + α
t∫
x
g(s)du(s), t ∈ (x, a+ b− x] ,
(1− α)
∫ t
b
g(s)du(s) + α
t∫
x
g(s)du(s), t ∈ (a+ b− x, b] .
Using integration by parts, we have the following identity:
b∫
a
Kg,u (t;x) df(t) =
x∫
a
(1− α) t∫
a
g(s)du(s) + α
t∫
x
g(s)du(s)
df(t)+
+
a+b−x∫
x
(1− α) t∫
(a+b)/2
g(s)du(s) + α
t∫
x
g(s)du(s)
df(t)+
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
898 M. W. ALOMARI
+
b∫
a+b−x
(1− α) t∫
b
g(s)du(s) + α
t∫
x
g(s)du(s)
df(t) =
= (1− α)
f(x) (a+b)/2∫
a
g(s)du(s) + f (a+ b− x)
b∫
(a+b)/2
g(s)du(s)
+
+α
f(a) x∫
a
g(s)du(s) + f(b)
b∫
x
g(s)du(s)
− b∫
a
f(t)g(t)du(t).
Using the fact that for a continuous function p : [a, b]→ R and a function ν : [a, b]→ R of bounded
variation, then the Riemann – Stieltjes integral
∫ b
a
p(t)dν(t) exists and one has the inequality
∣∣∣∣∣∣
b∫
a
p(t)dν(t)
∣∣∣∣∣∣ ≤ sup
t∈[a,b]
|p(t)|
b∨
a
(ν) . (2.2)
As f is of bounded variation on [a, b], by (2.2) we have∣∣∣∣∣∣∣(1− α)
f(x) (a+b)/2∫
a
g(s)du(s) + f (a+ b− x)
b∫
(a+b)/2
g(s)du(s)
+
+α
f(a) x∫
a
g(s)du(s) + f(b)
b∫
x
g(s)du(s)
− b∫
a
f(t)g(t)du(t)
∣∣∣∣∣∣ ≤
≤ sup
t∈[a,b]
|Kg,u (t;x)|
b∨
a
(f). (2.3)
Now, define the mappings p, q : [a, b]→ R given by
p1(t) := (1− α)
t∫
a
g(s)du(s) + α
t∫
x
g (s) du(s) t ∈ [a, x] ,
p2(t) := (1− α)
t∫
(a+b)/2
g(s)du(s) + α
t∫
x
g(s)du(s), t ∈ (x, a+ b− x] ,
p3(t) := (1− α)
t∫
b
g(s)du(s) + α
t∫
x
g (s) du(s), t ∈ (a+ b− x, b] ,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
NEW SHARP INEQUALITIES OF OSTROWSKI TYPE AND GENERALIZED TRAPEZOID TYPE . . . 899
for all α ∈ [0, 1], and x ∈ [a, b]. Since g is positive continuous and u is monotonic increasing on
[a, b] then the Riemann – Stieltjes integral
∫ b
a
g(s)du(s) exists and positive. Also, since the derivative
of the monotonic increasing function u is always positive, so that (gu′) (t) > 0 a.e., it follows that,
p′1(t), p
′
2(t), p
′
3(t) > 0, almost everywhere on their corresponding domains. Therefore, we have
sup
t∈[a,x]
|Kg,u (t;x)| = max
(1− α)
x∫
a
g(s)du(s), α
x∫
a
g(s)du(s)
=
=
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣]
x∫
a
g(s)du(s),
sup
t∈(x,a+b−x]
|Kg,u (t;x)| =
= max
(1− α)
(a+b)/2∫
x
g(s)du(s), α
(a+b)/2∫
x
g(s)du(s) +
a+b−x∫
(a+b)/2
g(s)du(s)
=
=
1
2
a+b−x∫
x
g(s)du(s) + (1− α)
∣∣∣∣∣∣∣
a+b−x∫
(a+b)/2
g(s)du(s)
∣∣∣∣∣∣∣
,
and
sup
t∈(a+b−x,b]
|Kg,u (t;x)| = max
(1− α)
b∫
x
g(s)du(s), α
b∫
x
g(s)du(s)
=
=
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣]
b∫
x
g(s)du(s).
Thus
sup
t∈[a,b]
|Kg,u (t;x)| =
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣]max
x∫
a
g(s)du(s),
b∫
x
g(s)du(s)
=
=
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣]
1
2
b∫
a
g(s)du(s) +
∣∣∣∣∣∣
x∫
a
g(s)du(s)− 1
2
b∫
a
g(s)du(s)
∣∣∣∣∣∣
. (2.4)
Therefore, by (2.3) and (2.4) we get (2.1). To prove that the constant
1
2
+
∣∣∣∣12 − α
∣∣∣∣ is best possible
for all α ∈ [0, 1], take u(t) = t for all t ∈ [a, b] and therefore, we refer to (1.1). Thus, the sharpness
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
900 M. W. ALOMARI
follows from (1.1) (consider f and g to be defined as in [19]). Hence, the proof is established and
we shall omit the details.
Corollary 2.1. In Theorem 2.1, choose α = 0, then we get∣∣∣∣∣∣f(x)
b∫
a
g(t)du(t)−
b∫
a
f(t)g(t)du(t)
∣∣∣∣∣∣ ≤
≤
1
2
b∫
a
g(t)du(t) +
∣∣∣∣∣∣
x∫
a
g(t)du(t)− 1
2
b∫
a
g(t)du(t)
∣∣∣∣∣∣
b∨
a
(f). (2.5)
A general weighted version of the above Ostrowski inequality for RS-integrals, may be deduced as
follows: ∣∣∣∣∣∣∣∣f(x)−
∫ b
a
f(t)g(t)du(t)∫ b
a
g(t)du(t)
∣∣∣∣∣∣∣∣ ≤
12 +
∣∣∣∣∣∣∣∣
∫ x
a
g(t)du(t)∫ b
a
g(t)du(t)
− 1
2
∣∣∣∣∣∣∣∣
b∨
a
(f) (2.6)
provided that g(t) ≥ 0, for almost every t ∈ [a, b] and
∫ b
a
g(t)du(t) 6= 0.
Remark 2.1. Choosing α = 1 in (2.1), then we get∣∣∣∣∣∣f(a)
x∫
a
g(s)du(s) + f(b)
b∫
x
g(s)du(s)−
b∫
a
f(t)g(t)du(t)
∣∣∣∣∣∣ ≤
≤
1
2
b∫
a
g(t)du(t) +
∣∣∣∣∣∣
x∫
a
g(t)du(t)− 1
2
b∫
a
g(t)du(t)
∣∣∣∣∣∣
b∨
a
(f), (2.7)
which is ‘the generalized trapezoid inequality for RS-integrals’.
Corollary 2.2. In Theorem 2.1, let g(t) = 1 for all t ∈ [a, b]. Then we have the inequality∣∣∣∣∣∣α [(u(x)− u(a)) f(a) + ((b)− u(x)) f(b)] + (1− α) [u(b)− u(a)] f(x)−
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] [u(b)− u(a)2
+
∣∣∣∣u(x)− u(a) + u(b)
2
∣∣∣∣] b∨
a
(f). (2.8)
The constant
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] is the best possible.
For instance,
If α = 0, then we get∣∣∣∣∣∣[u(b)− u(a)] f(x)−
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
[
u(b)− u(a)
2
+
∣∣∣∣u(x)− u(a) + (b)
2
∣∣∣∣] b∨
a
(f). (2.9)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
NEW SHARP INEQUALITIES OF OSTROWSKI TYPE AND GENERALIZED TRAPEZOID TYPE . . . 901
If α =
1
3
, then we have
∣∣∣∣∣∣13 {[u(x)− u(a)] f(a) + 2 [u(b)− u(a)] f(x) + [u(b)− u(x)] f(b)} −
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤ 2
3
[
u(b)− u(a)
2
+
∣∣∣∣u(x)− u(a) + u(b)
2
∣∣∣∣] b∨
a
(f). (2.10)
If α =
1
2
, then we obtain
∣∣∣∣∣∣12 {[u(x)− u(a)] f(a) + [u(b)− u(a)] f(x) + [u(b)− u(x)] f(b)} −
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤ 1
2
[
u(b)− u(a)
2
+
∣∣∣∣u(x)− u(a) + u(b)
2
∣∣∣∣] b∨
a
(f). (2.11)
If α = 1, then we get∣∣∣∣∣∣[u(x)− u(a)] f(a) + [u(b)− u(x)] f(b)−
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤
[
u(b)− u(a)
2
+
∣∣∣∣u(x)− u(a) + (b)
2
∣∣∣∣] b∨
a
(f). (2.12)
Proof. The results follow by Theorem 2.1. It remains to prove the sharpness of (2.8). Suppose
0 ≤ α ≤ 1
2
, assume that (2.8) holds with constant C1 > 0, i.e.,
∣∣∣∣∣∣α [(u(x)− u(a)) f(a) + ((b)− u(x)) f(b)]+ (1− α) [u(b)− u(a)] f(x)−
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤ C1
[
u(b)− u(a)
2
+
∣∣∣∣u(x)− u(a) + u(b)
2
∣∣∣∣] b∨
a
(f). (2.13)
Let f, u : [a, b]→ R be defined as follows u(t) = t and
f(t) =
0, t ∈ [a, b] \
{
a+ b
2
}
,
1
2
, t =
a+ b
2
,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
902 M. W. ALOMARI
which follows that
∨b
a(f) = 1 and
∫ b
a
f(t)du(t) = 0, setting x =
a+ b
2
it gives by (2.13)
(1− α) b− a
2
≤ C1
b− a
2
,
which proves that C1 ≥ 1− α, and therefore 1− α is the best possible for all 0 ≤ α ≤ 1
2
.
Now, suppose
1
2
≤ α ≤ 1 and assume that (2.8) holds with constant C2 > 0, i.e.,∣∣∣∣∣∣α [(u(x)− u(a)) f(a) + ((b)− u(x)) f(b)]+ (1− α) [u(b)− u(a)] f(x)−
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤ C2
[
u(b)− u(a)
2
+
∣∣∣∣u(x)− u(a) + u(b)
2
∣∣∣∣] b∨
a
(f). (2.14)
Let f, u : [a, b]→ R be defined as follows u(t) = t and
f(t) =
0, t ∈ (a, b] ,
1, t = a,
which follows that
∨b
a(f) = 1 and
∫ b
a
f(t)du(t) = 0, setting x =
a+ b
2
it gives by (2.14)
α
b− a
2
≤ C2
b− a
2
,
which proves that C2 ≥ α, and therefore α is the best possible for all
1
2
≤ α ≤ 1. Consequently, we
can conclude that the constant
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] is the best possible, for all α ∈ [0, 1].
Corollary 2.3. In (2.10), setting x =
a+ b
2
then we have the following Simpson-type inequality
for Riemann – Stieltjes integral:∣∣∣∣∣∣13
{[
u
(
a+ b
2
)
− u(a)
]
f(a) + 2 [u(b)− u(a)] f
(
a+ b
2
)
+
+
[
u(b)− u
(
a+ b
2
)]
f(b)
}
−
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤ 2
3
[
u(b)− u(a)
2
+
∣∣∣∣u(a+ b
2
)
− u(a) + u(b)
2
∣∣∣∣] b∨
a
(f). (2.15)
The constant
2
3
is the best possible.
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NEW SHARP INEQUALITIES OF OSTROWSKI TYPE AND GENERALIZED TRAPEZOID TYPE . . . 903
Remark 2.2. For recent three-point quadrature rules and related inequalities regarding Rie-
mann – Stieltjes integrals, the reader may refer to the work [2].
Corollary 2.4. In (2.8), let u(t) = t for all t ∈ [a, b], then we get∣∣∣∣∣∣α ((x− a) f(a) + (b− x) f(b)) + (1− α) (b− a) f(x)−
b∫
a
f(t)dt
∣∣∣∣∣∣ ≤
≤
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] [b− a2
+
∣∣∣∣x− a+ b
2
∣∣∣∣] b∨
a
(f). (2.16)
For x =
a+ b
2
, we have∣∣∣∣∣∣(b− a)
[
α
f(a) + f(b)
2
+ (1− α) f
(
a+ b
2
)]
−
b∫
a
f (t) dt
∣∣∣∣∣∣ ≤
≤
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] b− a2
b∨
a
(f). (2.17)
Remark 2.3. Under the assumptions of Theorem 2.1, a weighted generalization of Mont-
gomery’s type identity for Riemann – Stieltjes integrals may be deduced as follows:
f(x) =
1∫ b
a
g(s)du(s)
b∫
a
Kg,u (t;x) df(t) +
1∫ b
a
g(s)du(s)
b∫
a
f(t)g(t)du(t),
for all x ∈ [a, b], where
Kg,u (t;x) :=
∫ t
a
g(s)du(s), t ∈ [a, x],∫ t
b
g(s)du(s), t ∈ (x, b].
Provided that
∫ b
a
g(s)du(s) 6= 0.
3. On L-Lipschitz integrators.
Theorem 3.1. Let g be as in Theorem 2.1. Let u : [a, b] → [0,∞) be of bounded variation on
[a, b]. If f : [a, b]→ R is L-Lipschitzian on [a, b], then for any x ∈ [a, b] and α ∈ [0, 1], we have∣∣∣∣∣∣α
f(a) x∫
a
g(s)du(s) + f(b)
b∫
x
g(s)du(s)
+
+(1− α) f(x)
b∫
a
g(s)du(s)−
b∫
a
f(t)g(t)du(t)
∣∣∣∣∣∣ ≤
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904 M. W. ALOMARI
≤ Lmax
{
(x− a) sup
t∈[a,x]
{M(t)}, (b− x) sup
t∈[x,b]
{N(t)}
}
b∨
a
(u), (3.1)
where
M(t) := max
{
(1− α) sup
s∈[a,t]
|g(s)|, α sup
s∈[t,x]
|g(s)|
}
and
N(t) := max
{
(1− α) sup
s∈[t,b]
|g(s)|, α sup
s∈[t,x]
|g(s)|
}
.
Proof. By Theorem 2.1, we have the identity
b∫
a
Kg,u (t;x) df(t) =
= α
f(a) x∫
a
g(s)du(s) + f(b)
b∫
x
g(s)du(s)
+
+(1− α) f(x)
b∫
a
g(s)du(s)−
b∫
a
f(t)g(t)du(t).
Using the fact that for a Riemann integrable function p : [c, d] → R and L-Lipschitzian function
ν : [c, d]→ R, the inequality one has the inequality∣∣∣∣∣∣
d∫
c
p(t)dν(t)
∣∣∣∣∣∣ ≤ L
d∫
c
|p(t)| dt. (3.2)
As f is L-Lipschitz mapping on [a, b], by (3.2) we have∣∣∣∣∣∣
b∫
a
Kg,u (t;x) df(t)
∣∣∣∣∣∣ ≤ L
b∫
a
|Kg,u (t;x)| dt = L
x∫
a
|p(t)| dt+
b∫
x
|q(t)| dt
. (3.3)
However, as u is of bounded variation on [a, b] and g is continuous, by (2.2) we obtain
|p(t)| ≤ (1− α)
∣∣∣∣∣∣
t∫
a
g(s)du(s)
∣∣∣∣∣∣+ α
∣∣∣∣∣∣
t∫
x
g(s)du(s)
∣∣∣∣∣∣ ≤
≤ (1− α) sup
s∈[a,t]
|g(s)|
t∨
a
(u) + α sup
s∈[t,x]
|g(s)|
x∨
t
(u) ≤
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NEW SHARP INEQUALITIES OF OSTROWSKI TYPE AND GENERALIZED TRAPEZOID TYPE . . . 905
≤ max
{
(1− α) sup
s∈[a,t]
|g(s)| , α sup
s∈[t,x]
|g(s)|
}
x∨
a
(u) :=
:=M(t)
x∨
a
(u). (3.4)
Similarly, we get
|q(t)| ≤ max
{
(1− α) sup
s∈[t,b]
|g(s)| , α sup
s∈[t,x]
|g(s)|
}
b∨
x
(u) := N(t)
b∨
x
(u). (3.5)
Thus, by (3.3) – (3.5), we have∣∣∣∣∣∣
b∫
a
Kg,u (t;x) df(t)
∣∣∣∣∣∣ ≤ L
x∫
a
|p(t)| dt+
b∫
x
|q(t)| dt
≤
≤ L
x∫
a
M(t)dt
x∨
a
(u) +
b∫
x
N(t)dt
b∨
x
(u)
≤
≤ L
[
(x− a) sup
t∈[a,x]
{M(t)}
x∨
a
(u) + (b− x) sup
t∈[x,b]
{N(t)}
b∨
x
(u)
]
≤
≤ Lmax
{
(x− a) sup
t∈[a,x]
{M(t)}, (b− x) sup
t∈[x,b]
{N(t)}
}
b∨
a
(u),
which gives the result.
Remark 3.1. In Theorem 3.1, if g(t) = 1 for all t ∈ [a, b]. Then
M(t) = N(t) =
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣], for all t ∈ [a, b].
Corollary 3.1. In Theorem 3.1, let g(t) = 1 for all t ∈ [a, b]. Then, we have the inequality∣∣∣∣∣∣α [(u(x)− u(a)) f(a) + ((b)− u(x)) f(b)] + (1− α) [u(b)− u(a)] f(x)−
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤ L
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] [b− a2
+
∣∣∣∣x− a+ b
2
∣∣∣∣] b∨
a
(u). (3.6)
The constant
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] is the best possible.
For instance,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
906 M. W. ALOMARI
If α = 0, then we get∣∣∣∣∣∣[u(b)− u(a)] f(x)−
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤ L
[
b− a
2
+
∣∣∣∣x− a+ b
2
∣∣∣∣] b∨
a
(u). (3.7)
If α =
1
3
, then we obtain
∣∣∣∣∣∣13 {[u(x)− u(a)] f(a) + 2 [u (b)− u(a)] f(x) + [u(b)− u(x)] f(b)} −
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤ 2
3
L
[
b− a
2
+
∣∣∣∣x− a+ b
2
∣∣∣∣] b∨
a
(u). (3.8)
If α =
1
2
, then we have
∣∣∣∣∣∣12 {[u(x)− u(a)] f(a) + [u(b)− u(a)] f(x) + [u(b)− u(x)] f(b)} −
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤ 1
2
L
[
b− a
2
+
∣∣∣∣x− a+ b
2
∣∣∣∣] b∨
a
(u). (3.9)
If α = 1, then we get∣∣∣∣∣∣[u(x)− u(a)] f(a) + [u(b)− u(x)] f(b)−
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤ L
[
b− a
2
+
∣∣∣∣x− a+ b
2
∣∣∣∣] b∨
a
(u). (3.10)
Proof. The results follow by Theorem 3.1. It remains to prove the sharpness of (3.6). Suppose
0 ≤ α ≤ 1
2
, assume that (3.6) holds with constant C1 > 0, i.e.,
∣∣∣∣∣∣α [(u(x)− u(a)) f(a) + ((b)− u(x)) f(b)]+
+ (1− α) [u(b)− u(a)] f(x)−
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
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NEW SHARP INEQUALITIES OF OSTROWSKI TYPE AND GENERALIZED TRAPEZOID TYPE . . . 907
≤ LC1
[
b− a
2
+
∣∣∣∣x− a+ b
2
∣∣∣∣] b∨
a
(u). (3.11)
Let f, u : [a, b]→ R be defined as follows f(t) = t− b and
u(t) =
0, t ∈ [a, b),
1, t = b.
Therefore, f is L-Lipschitz with L = 1 and
∨b
a(u) = 1 and
∫ b
a
f(t)du(t) = 0, setting x =
a+ b
2
it
gives by (3.11)
(1− α) b− a
2
≤ C1
b− a
2
,
which proves that C1 ≥ 1− α, and therefore 1− α is the best possible for all 0 ≤ α ≤ 1
2
.
Now, suppose
1
2
≤ α ≤ 1 and assume that (3.6) holds with constant C2 > 0, i.e.,
∣∣∣∣∣∣α [(u(x)− u(a)) f(a) + ((b)− u(x)) f(b)]+ (1− α) [u(b)− u(a)] f(x)−
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤ LC2
[
b− a
2
+
∣∣∣∣x− a+ b
2
∣∣∣∣] b∨
a
(u). (3.12)
Let f, u : [a, b]→ R be defined as follows f(t) = t− a and
u(t) =
0, t ∈ [a, b] \
{
a+ b
2
}
,
1
2
, t =
a+ b
2
.
Therefore, f is L-Lipschitz with L = 1 and
∨b
a(u) = 1 and
∫ b
a
f(t)du(t) = 0, setting x =
a+ b
2
it
gives by (3.12)
α
b− a
2
≤ C2
b− a
2
,
which proves that C2 ≥ α, and therefore α is the best possible for all
1
2
≤ α ≤ 1. Consequently, we
can conclude that the constant
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] is the best possible, for all α ∈ [0, 1].
Corollary 3.2. In (3.8), choosing x =
a+ b
2
, then we have the following Simpson-type inequal-
ity for RS-integrals:
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
908 M. W. ALOMARI∣∣∣∣∣∣13
{[
u
(
a+ b
2
)
− u(a)
]
f(a) + 2 [u(b)− u(a)] f
(
a+ b
2
)
+
+
[
u(b)− u
(
a+ b
2
)]
f(b)
}
−
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤ 1
3
L (b− a)
b∨
a
(u). (3.13)
The constant
1
3
is the best possible.
Corollary 3.3. In (3.6), let u(t) = t for all t ∈ [a, b], then we get∣∣∣∣∣∣α ((x− a) f(a) + (b− x) f(b)) + (1− α) (b− a) f(x)−
b∫
a
f(t)dt
∣∣∣∣∣∣ ≤
≤ L (b− a)
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] [b− a2
+
∣∣∣∣x− a+ b
2
∣∣∣∣]. (3.14)
For x =
a+ b
2
, we have∣∣∣∣∣∣(b− a)
[
α
f(a) + f(b)
2
+ (1− α) f
(
a+ b
2
)]
−
b∫
a
f (t) dt
∣∣∣∣∣∣ ≤
≤ L
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] (b− a)22
. (3.15)
4. On monotonic nondecreasing integrators.
Theorem 4.1. Let g, u be as in Theorem 3.1. If f : [a, b] → R is monotonic nondecreasing on
[a, b], then for any x ∈ [a, b] and α ∈ [0, 1], we have∣∣∣∣∣∣α
f(a) x∫
a
g(s)du(s) + f(b)
b∫
x
g(s)du(s)
+
+(1− α) f(x)
b∫
a
g(s)du(s)−
b∫
a
f(t)g(t)du(t)
∣∣∣∣∣∣ ≤
≤ sup
t∈[a,x]
{M(t)} [f(x)− f(a)]
x∨
a
(u) + sup
t∈[x,b]
{N(t)} [f(b)− f(x)]
b∨
x
(u), (4.1)
where M(t) and N(t) are defined in Theorem 3.1.
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NEW SHARP INEQUALITIES OF OSTROWSKI TYPE AND GENERALIZED TRAPEZOID TYPE . . . 909
Proof. Using the identity
α
f(a) x∫
a
g(s)du(s) + f(b)
b∫
x
g(s)du(s)
+
+(1− α) f(x)
b∫
a
g(s)du(s)−
b∫
a
f(t)g(t)du(t) =
=
b∫
a
Kg,u (t;x) df(t).
It is well-known that for a monotonic nondecreasing function ν : [a, b]→ R and continuous function
p : [a, b]→ R, one has the inequality∣∣∣∣∣∣
b∫
a
p(t)dν(t)
∣∣∣∣∣∣ ≤
b∫
a
|p(t)| dν(t). (4.2)
As f is monotonic nondecreasing on [a, b], by (4.2) we have∣∣∣∣∣∣
b∫
a
Kg,u (t;x) df(t)
∣∣∣∣∣∣ ≤
b∫
a
|Kg,u (t;x)| df(t) =
=
x∫
a
|p(t)| df(t) +
b∫
x
|q(t)| df(t). (4.3)
Now, as u is of bounded variation on [a, b] and g is continuous, by (3.4), (3.5) we obtain
|p(t)| ≤M(t)
x∨
a
(u),
|q(t)| ≤ N(t)
b∨
x
(u).
(4.4)
Thus, by (4.3) and (4.4), we get∣∣∣∣∣∣
b∫
a
Kg,u (t;x) df(t)
∣∣∣∣∣∣ ≤
x∫
a
|p(t)| df(t) +
b∫
x
|q(t)| df(t) ≤
≤
x∫
a
M(t)df(t)
x∨
a
(u) +
b∫
x
N(t)df(t)
b∨
x
(u) ≤
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910 M. W. ALOMARI
≤ sup
t∈[a,x]
{M(t)} [f(x)− f(a)]
x∨
a
(u) + sup
t∈[x,b]
{N(t)} [f(b)− f(x)]
b∨
x
(u),
which gives the result.
Corollary 4.1. In Theorem 4.1, let g(t) = 1 for all t ∈ [a, b]. Then, we have the inequality∣∣∣∣∣∣α [(u(x)− u(a)) f(a) + ((b)− u(x)) f(b)]+ (1− α) [u(b)− u(a)] f(x)−
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣]
{
[f(x)− f(a)]
x∨
a
(u) + [f(b)− f(x)]
b∨
x
(u)
}
≤
≤
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣][f(b)− f(a)2
+
∣∣∣∣f(x)− f(a) + f(b)
2
∣∣∣∣] b∨
a
(u). (4.5)
For the last inequality, the constant
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] is the best possible.
For instance,
If α = 0, then we have ∣∣∣∣∣∣[u(b)− u(a)] f(x)−
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤ [f(x)− f(a)]
x∨
a
(u) + [f(b)− f (x)]
b∨
x
(u) ≤
≤
[
f(b)− f(a)
2
+
∣∣∣∣f(x)− f(a) + f (b)
2
∣∣∣∣] b∨
a
(u). (4.6)
If α =
1
3
, then we get
∣∣∣∣∣∣13 {[u(x)− u(a)] f(a) + 2 [u (b)− u(a)] f(x) + [u(b)− u(x)] f(b)} −
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤ 2
3
{
[f(x)− f(a)]
x∨
a
(u) + [f(b)− f(x)]
b∨
x
(u)
}
≤
≤ 2
3
[
f(b)− f(a)
2
+
∣∣∣∣f(x)− f(a) + f(b)
2
∣∣∣∣] b∨
a
(u). (4.7)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
NEW SHARP INEQUALITIES OF OSTROWSKI TYPE AND GENERALIZED TRAPEZOID TYPE . . . 911
If α =
1
2
, then we obtain∣∣∣∣∣∣12 {[u(x)− u(a)] f(a) + [u(b)− u(a)] f(x) + [u(b)− u(x)] f(b)} −
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤ 1
2
{
[f(x)− f(a)]
x∨
a
(u) + [f(b)− f(x)]
b∨
x
(u)
}
≤
≤ 1
2
[
f(b)− f(a)
2
+
∣∣∣∣f(x)− f(a) + f(b)
2
∣∣∣∣] b∨
a
(u). (4.8)
If α = 1, then we have∣∣∣∣∣∣[u(x)− u(a)] f(a) + [u(b)− u(x)] f(b)−
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤ [f(x)− f(a)]
x∨
a
(u) + [f(b)− f (x)]
b∨
x
(u) ≤
≤
[
f(b)− f(a)
2
+
∣∣∣∣f(x)− f(a) + f (b)
2
∣∣∣∣] b∨
a
(u). (4.9)
Proof. The results follow by Theorem 4.1. It remains to prove the sharpness of (4.5). Suppose
0 ≤ α ≤ 1
2
, assume that (4.5) holds with constant C1 > 0, i.e.,∣∣∣∣∣∣α [(u(x)− u(a)) f(a) + ((b)− u(x)) f(b)] +
+ (1− α) [u(b)− u(a)] f(x)−
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤ C1
[
f(b)− f(a)
2
+
∣∣∣∣f(x)− f(a) + f (b)
2
∣∣∣∣] b∨
a
(u). (4.10)
Let f, u : [a, b]→ R be defined as follows:
f(t) =
−1, t = a,
0, t = (a, b],
and
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
912 M. W. ALOMARI
u(t) =
0, t ∈ [a, b) ,
1, t = b.
Therefore, f is monotonic nondecreasing on [a, b] and
∨b
a(u) = 1 and
∫ b
a
f(t)du(t) = 0, setting
x = a it gives by (4.10) that 1 − α ≤ C1, and which proves that 1 − α is the best possible for all
0 ≤ α ≤ 1
2
.
Now, suppose
1
2
≤ α ≤ 1 and assume that (4.5) holds with constant C2 > 0, i.e.,∣∣∣∣∣∣α [(u(x)− u(a)) f(a) + ((b)− u(x)) f(b)] +
+ (1− α) [u(b)− u(a)] f(x)−
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤ C2
[
f(b)− f(a)
2
+
∣∣∣∣f(x)− f(a) + f (b)
2
∣∣∣∣] b∨
a
(u). (4.11)
Let f, u : [a, b] → R be defined as f(t) as above, and u(t) = t, which follows that
∨b
a(u) = b − a,
and
∫ b
a
f(t)du(t) = 0, setting x = b it gives by (4.11) α ≤ C2, and therefore α is the best possible
for all
1
2
≤ α ≤ 1. Consequently, we can conclude that the constant
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] is the best
possible, for all α ∈ [0, 1].
Corollary 4.2. In (4.7), choosing x =
a+ b
2
, then we have the following Simpson-type inequal-
ity for RS-integrals:∣∣∣∣∣∣13
{[
u
(
a+ b
2
)
− u(a)
]
f(a) + 2 [u(b)− u(a)] f
(
a+ b
2
)
+
+
[
u(b)− u
(
a+ b
2
)]
f(b)
}
−
b∫
a
f(t)du(t)
∣∣∣∣∣∣ ≤
≤ 2
3
[
f
(
a+ b
2
)
− f(a)
] (a+b)/2∨
a
(u) +
[
f(b)− f
(
a+ b
2
)] b∨
(a+b)/2
(u)
≤
≤ 2
3
[
f(b)− f(a)
2
+
∣∣∣∣f (a+ b
2
)
− f(a) + f(b)
2
∣∣∣∣] b∨
a
(u). (4.12)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
NEW SHARP INEQUALITIES OF OSTROWSKI TYPE AND GENERALIZED TRAPEZOID TYPE . . . 913
For the last inequality, the constant
2
3
is the best possible.
Corollary 4.3. In (4.5), let u(t) = t for all t ∈ [a, b], then we get∣∣∣∣∣∣α ((x− a) f(a) + (b− x) f(b)) + (1− α) (b− a) f(x)−
b∫
a
f(t)dt
∣∣∣∣∣∣ ≤
≤
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] {(x− a) [f (x)− f(a)] + (b− x) [f(b)− f(x)]} ≤
≤
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] [f(b)− f(a)2
+
∣∣∣∣f(x)− f(a) + f(b)
2
∣∣∣∣] (b− a) . (4.13)
For x =
a+ b
2
, we have
∣∣∣∣∣∣(b− a)
[
α
f(a) + f(b)
2
+ (1− α) f
(
a+ b
2
)]
−
b∫
a
f (t) dt
∣∣∣∣∣∣ ≤
≤ 1
2
(b− a)
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] [f (b)− f(a)] ≤
≤
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] [f(b)− f(a)2
+
∣∣∣∣f (a+ b
2
)
− f(a) + f(b)
2
∣∣∣∣] (b− a) . (4.14)
Remark 4.1. We give an attention to the interested reader, is that, in Theorems 2.1, 3.1, 4.1,
one may observe various new inequalities by replacing the assumptions on u, e.g. to be of bounded
variation, Lu-Lipschitz or monotonic nondecreasing on [a, b], which therefore gives in some cases
the ‘dual’ of the above obtained inequalities.
It remains to mention that, in Theorem 3.1, and according to the assumptions on u one may
observe several estimations for the functions p(t) and q(t) which therefore gives different functions
M(t) and N(t).
Remark 4.2. In Theorems 2.1, 3.1, 4.1, a different result(s) in terms of Lp norms may be stated
by applying the well-known Hölder integral inequality, by noting that
∣∣∣∣∣∣
d∫
c
g(s)du(s)
∣∣∣∣∣∣ ≤ q
√
u (d)− u (c)× p
√√√√√ d∫
c
|g(s)|p du (s),
where p > 1,
1
p
+
1
q
= 1.
Remark 4.3. One can point out some results for the Riemann integral of a product, in terms of
L1-, Lp-, and L∞-norms by using a similar argument considered in [12] (see also [1, 2]).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
914 M. W. ALOMARI
5. Applications to Ostrowski generalized trapezoid quadrature formula for RS-integrals.
Let In : a = x0 < x1 < s < xn = b be a division of the interval [a, b]. Define the general Riemann –
Stieltjes sum
S (f, u, In, ξ) =
n−1∑
i=0
α {[u(ξi)− u(xi)] f(xi) + [u(xi+1)− u(ξi)] f(xi+1)}+
+(1− α) [u(xi+1)− u(xi)] f (ξi) . (5.1)
In the following, we establish an upper bound for the error approximation of the Riemann – Stieltjes
integral
∫ b
a
f(t)du(t) by its Riemann – Stieltjes sum S (f, u, In, ξ) . As a sample we apply the in-
equality (2.8).
Theorem 5.1. Under the assumptions of Corollary 2.2, we have
b∫
a
f(t)du(t) = S (f, u, In, ξ) +R (f, u, In, ξ) ,
where S (f, u, In, ξ) is given in (5.1) and the remainder R (f, u, In, ξ) satisfies the bound
|R (f, u, In, ξ)| ≤
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] [u(b)− u(a)] b∨
a
(f). (5.2)
Proof. Applying Corollary 2.2 on the intervals [xi, xi+1], we may state that∣∣∣∣∣∣α {[u(ξi)− u(xi)] f(xi) + [u(xi+1)− u(ξi)] f(xi+1)} +
+(1− α) [u (xi+1)− u(xi)] f (ξi)−
xi+1∫
xi
f(t)du(t)
∣∣∣∣∣∣ ≤
≤
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] [u(xi+1)− u(xi)
2
+
∣∣∣∣u(ξi)− u(xi) + u(xi+1)
2
∣∣∣∣] xi+1∨
xi
(f)
for all i ∈ {0, 1, 2, s, n− 1}.
Summing the above inequality over i from 0 to n−1 and using the generalized triangle inequality,
we deduce
|R (f, u, In, ξ)| =
n−1∑
i=0
∣∣∣∣∣∣α {[u(ξi)− u(xi)] f(xi) + [u (xi+1)− u(ξi)] f (xi+1)} +
+(1− α) [u (xi+1)− u(xi)] f (ξi)−
xi+1∫
xi
f(t)du(t)
∣∣∣∣∣∣ ≤
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
NEW SHARP INEQUALITIES OF OSTROWSKI TYPE AND GENERALIZED TRAPEZOID TYPE . . . 915
≤
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] n−1∑
i=0
[
u (xi+1)− u(xi)
2
+
∣∣∣∣u (ξi)− u(xi) + u (xi+1)
2
∣∣∣∣] xi+1∨
xi
(f) ≤
≤
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣]
[
n−1∑
i=0
u (xi+1)− u(xi)
2
+
n−1∑
i=0
∣∣∣∣u(ξi)− u(xi) + u(xi+1)
2
∣∣∣∣
]
n−1∑
i=0
xi+1∨
xi
(f) ≤
≤
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣]
[
u(b)− u(a)
2
+ sup
i=0,1,...,n−1
∣∣∣∣u(ξi)− u(xi) + u (xi+1)
2
∣∣∣∣
]
b∨
a
(f) ≤
≤
[
1
2
+
∣∣∣∣12 − α
∣∣∣∣] [u(b)− u(a)] b∨
a
(f).
Since
sup
i=0,1,...,n−1
∣∣∣∣u(ξi)− u(xi) + u (xi+1)
2
∣∣∣∣ ≤ sup
i=0,1,...,n−1
u(xi+1)− u(xi)
2
=
u(b)− u(a)
2
and
n−1∑
i=0
xi+1∨
xi
(f) =
b∨
a
(f),
which completes the proof.
Remark 5.1. One may use the remaining inequalities in Section 2, to obtain other bounds for
R (f, u, In, ξ). We shall omit the details.
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916 M. W. ALOMARI
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Received 31.05.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
|
| id | umjimathkievua-article-2476 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:24:08Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/cd/ae148537f5f2a5a6161a94e8b4c167cd.pdf |
| spelling | umjimathkievua-article-24762020-03-18T19:16:28Z New Sharp Ostrowski-type Inequalities and Generalized Trapezoid-type Inequalities for Riemann–Stieltjes Integrals and their Applications Нові точні неперервності типу Островського та типу узагальненого трапецоїда для інтегралів Рімана - Стільтьєса та їх застосування Alomari, M. W. Аломарі, М. В. We prove new sharp weighted generalizations of Ostrowski-type and generalized trapezoid-type inequalities for Riemann–Stieltjes integrals. Several related inequalities are deduced and investigated. New Simpson-type inequalities are obtained for the \( \mathcal{R}\mathcal{S} \) -integral. Finally, as an application, we estimate the error of a general quadrature rule for the \( \mathcal{R}\mathcal{S} \) -integral via the Ostrowski–generalized-trapezoid-quadrature formula. Доведено нові точні зважені узагальнення нєрівностєй типу Островського та типу узагальненого трапецоїда для iнтегралiв Рімана-Стільтьєса. Отримано та досліджено кілька близьких нерівностей. Отримано нові нерівності типу Сімпсона для \( \mathcal{R}\mathcal{S} \) -інтеграла. Як застосування наведено оцінку похибки загального правила квадратур для \( \mathcal{R}\mathcal{S} \) -інтеграла із використанням квадратурної формули Островського — узагальненого трапецоїда. Institute of Mathematics, NAS of Ukraine 2013-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2476 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 7 (2013); 894–916 Український математичний журнал; Том 65 № 7 (2013); 894–916 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2476/1718 https://umj.imath.kiev.ua/index.php/umj/article/view/2476/1719 Copyright (c) 2013 Alomari M. W. |
| spellingShingle | Alomari, M. W. Аломарі, М. В. New Sharp Ostrowski-type Inequalities and Generalized Trapezoid-type Inequalities for Riemann–Stieltjes Integrals and their Applications |
| title | New Sharp Ostrowski-type Inequalities and Generalized Trapezoid-type Inequalities for Riemann–Stieltjes Integrals and their Applications |
| title_alt | Нові точні неперервності типу Островського та типу узагальненого трапецоїда для інтегралів Рімана - Стільтьєса та їх застосування |
| title_full | New Sharp Ostrowski-type Inequalities and Generalized Trapezoid-type Inequalities for Riemann–Stieltjes Integrals and their Applications |
| title_fullStr | New Sharp Ostrowski-type Inequalities and Generalized Trapezoid-type Inequalities for Riemann–Stieltjes Integrals and their Applications |
| title_full_unstemmed | New Sharp Ostrowski-type Inequalities and Generalized Trapezoid-type Inequalities for Riemann–Stieltjes Integrals and their Applications |
| title_short | New Sharp Ostrowski-type Inequalities and Generalized Trapezoid-type Inequalities for Riemann–Stieltjes Integrals and their Applications |
| title_sort | new sharp ostrowski-type inequalities and generalized trapezoid-type inequalities for riemann–stieltjes integrals and their applications |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2476 |
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