Simpson-type inequalities for geometrically relative convex functions
We consider a class of geometrically relative convex functions and deduce several new integral inequalities of Simpson’s type via geometrically relative convex functions. The ideas and techniques used in the paper may stimulate further research in this area.
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Institute of Mathematics, NAS of Ukraine
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| author | Awan, M. U. Noor, K. I. Noor, M. A. Аван, М. У. Нур, К. І. Нур, М. А. |
| author_facet | Awan, M. U. Noor, K. I. Noor, M. A. Аван, М. У. Нур, К. І. Нур, М. А. |
| author_sort | Awan, M. U. |
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| datestamp_date | 2019-12-05T09:20:38Z |
| description | We consider a class of geometrically relative convex functions and deduce several new integral inequalities of Simpson’s
type via geometrically relative convex functions. The ideas and techniques used in the paper may stimulate further research
in this area. |
| first_indexed | 2026-03-24T02:09:07Z |
| format | Article |
| fulltext |
UDC 517.5
M. A. Noor, K. I. Noor (COMSATS Inst. Inform. Technology, Islamabad, Pakistan),
M. U. Awan (GC Univ., Faisalabad, Pakistan)
SIMPSON-TYPE INEQUALITIES FOR GEOMETRICALLY RELATIVE
CONVEX FUNCTIONS
НЕРIВНОСТI ТИПУ СIМПСОНА ДЛЯ ГЕОМЕТРИЧНО ВIДНОСНИХ
ОПУКЛИХ ФУНКЦIЙ
We consider a class of geometrically relative convex functions and deduce several new integral inequalities of Simpson’s
type via geometrically relative convex functions. The ideas and techniques used in the paper may stimulate further research
in this area.
Розглянуто клас геометрично вiдносних опуклих функцiй та отримано кiлька нових iнтегральних нерiвностей типу
Сiмпсона в термiнах геометрично вiдносних опуклих функцiй. Iдеї i технiка, що використовуються в роботi, можуть
стимулювати подальшi дослiдження в данiй областi.
1. Introduction. Let f : I = [a, b] \subseteq \BbbR \rightarrow \BbbR be a four time continuously differentiable on I\circ ,
where I\circ is the interior of I and \| f (4)\| \infty < \infty . Then following inequality is known as Simpson’s
inequality in the literature:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 13
\biggl[
f(a) + f(b)
2
+ 2f
\biggl(
a+ b
2
\biggr) \biggr]
- 1
b - a
b\int
a
f(x)dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq 1
2880
\| f (4)\| \infty (b - a)4. (1.1)
For useful details on Simpson’s type of integral inequalities (see [1 – 3, 5, 13, 16, 17]). Convexity
plays an important role in different fields of pure and applied sciences. Due to its significance,
many researchers have paid much attention to this subject by investigating its various properties.
Consequently, the concept of convexity has been extended and generalized in different directions
using novel and innovative ideas (see [4, 6 – 12, 14, 15, 17]).
Noor et al. [10] introduced and investigated the concept of geometrically relative convex func-
tions, which also contains the class of relative convex functions as special case.
In this paper, we consider the class of geometrically relative convex functions and derive several
new Simpson’s type of integral inequalities. This is the main motivation of this paper.
2. Preliminaries. In this section, we recall some previously known concepts.
Definition 2.1 [10]. Let \scrG \subseteq (0,\infty ). Then \scrG is said to be geometrically relative convex set, if
there exists an arbitrary function g : \BbbR n \rightarrow \BbbR n such that
(g(x))t(g(y))1 - t \in \scrG \forall g(x), g(y) \in \scrG , t \in [0, 1].
Using AM - GM inequality, we have
(g(x))t(g(y))1 - t \leq tg(x) + (1 - t)g(y) \forall g(x), g(y) \in \scrG , t \in [0, 1].
Definition 2.2 [10]. A function f : \scrG \rightarrow \BbbR
\bigl(
on subintervals of (0,\infty )
\bigr)
is said to be geometri-
cally relative convex function (GG-relative convex function) if there exists an arbitrary function g :
\BbbR n \rightarrow \BbbR n such that
f((g(x))t(g(y))1 - t) \leq (f(g(x)))t(f(g(y)))1 - t \forall g(x), g(y) \in \scrG , t \in [0, 1]. (2.1)
c\bigcirc M. A. NOOR, K. I. NOOR, M. U. AWAN, 2018
992 ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
SIMPSON-TYPE INEQUALITIES FOR GEOMETRICALLY RELATIVE CONVEX FUNCTIONS 993
From (2.1), it follows that
\mathrm{l}\mathrm{o}\mathrm{g} f((g(x))t(g(y))1 - t) \leq t \mathrm{l}\mathrm{o}\mathrm{g} f(g(x)) + (1 - t) \mathrm{l}\mathrm{o}\mathrm{g} f(g(y)) \forall g(x), g(y) \in \scrG , t \in [0, 1].
Using AM - GM inequality, we have
f((g(x))t(g(y))1 - t) \leq (f(g(x)))t(f(g(y)))1 - t \leq
\leq tf(g(x)) + (1 - t)f(g(y)).
Thus it follows that every geometrically relative convex function (GG-relative convex function) is
also GA-relative convex function, but the converse is not true.
For t =
1
2
in (2.1), we have Jensen type of geometrically relative convex functions. That is
f
\Bigl( \sqrt{}
g(x)g(y)
\Bigr)
\leq
\sqrt{}
f(g(x))f(g(y)).
Definition 2.3 [10]. Let I be a subinterval of (0,\infty ). Then f is geometrically relative convex
function if and only if \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1 1 1
\mathrm{l}\mathrm{o}\mathrm{g} g(a) \mathrm{l}\mathrm{o}\mathrm{g} g(x) \mathrm{l}\mathrm{o}\mathrm{g} g(b)
\mathrm{l}\mathrm{o}\mathrm{g} f(g(a)) \mathrm{l}\mathrm{o}\mathrm{g} f(g(x)) \mathrm{l}\mathrm{o}\mathrm{g} f(g(b))
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \geq 0,
where g(a) \leq g(x) \leq g(b).
One can easily show that the following are equivalent:
(1) f is geometrically relative convex function on geometrically relative convex set;
(2) f(g(a))log(g(b))f(g(x))log(g(a))f(g(b))log(g(x)) \geq f(g(a))log(g(x))f(g(x))log(g(b)) \times
\times f(g(b))log(g(a)), where g(x) = g(a)tg(b)1 - t and t \in [0, 1].
Definition 2.4 [10]. A function f : \scrG \rightarrow \BbbR (on subintervals of (0,\infty )) is said to be GA-relative
convex function, if there exists an arbitrary function g : \BbbR n \rightarrow \BbbR n such that
f((g(x))t(g(y))1 - t) \leq tf(g(x)) + (1 - t)f(g(y)) \forall g(x), g(y) \in \scrG , t \in [0, 1]. (2.2)
From Definitions 2.3 and 2.4, it follows that GG =\Rightarrow GA, but the converse is not true.
3. Main results. In this section, we prove our main results.
Lemma 3.1. For g(a), g(b) \in \scrG and t \in [0, 1], if g(a) < g(b), then
(g(a))1 - t(g(b))t \leq (1 - t)g(a) + tg(b).
Essentially using the technique of [2], one can prove following result.
Lemma 3.2. Let f : I \subset \BbbR \rightarrow \BbbR be an absolutely continuous function on I\circ , where g(a),
g(b) \in I with g(a) < g(b). Then following equality holds:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1
6
\biggl[
f(g(a)) + 4f
\biggl(
g(a) + g(b)
2
\biggr)
+ f(g(b))
\biggr]
- 1
g(b) - g(a)
g(b)\int
g(a)
f(g(x))dg(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
994 M. A. NOOR, K. I. NOOR, M. U. AWAN
= (g(b) - g(a))
1\int
0
\mu (t)f \prime ((1 - t)g(a) + tg(b))dt,
where
\mu (t) =
\left\{
t - 1
6
, if t \in
\biggl[
0,
1
2
\biggr)
,
t - 5
6
, if t \in
\biggl[
1
2
, 1
\biggr]
.
Now using Lemmas 3.1 and 3.2 we prove our main results.
Theorem 3.1. Let f : I \subset \BbbR \rightarrow \BbbR be a differentiable function on I\circ , where g(a), g(b) \in I with
g(a) < g(b). If f \prime \in L[g(a), g(b)] and | f \prime | is monotonically decreasing and geometrically relative
convex function, then following inequality holds:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1
6
\biggl[
f(g(a)) + 4f
\biggl(
g(a) + g(b)
2
\biggr)
+ f(g(b))
\biggr]
- 1
g(b) - g(a)
g(b)\int
g(a)
f(g(x))dg(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq (g(b) - g(a))| f \prime (g(a))| \{ F1(t, w) + F2(t, w)\} ,
where
F1(t, w) =
1
2\int
0
\bigm| \bigm| \bigm| \bigm| t - 1
6
\bigm| \bigm| \bigm| \bigm| wtdt =
- 6 + 12w
1
6 - 6
\surd
w - \mathrm{l}\mathrm{n}w + 2
\surd
w \mathrm{l}\mathrm{n}w
6 \mathrm{l}\mathrm{n}w2
and
F2(t, w) =
1\int
1
2
\bigm| \bigm| \bigm| \bigm| 16 - t
\bigm| \bigm| \bigm| \bigm| wtdt =
6
\surd
w - 6w - 2
\surd
w \mathrm{l}\mathrm{n}w + 5w \mathrm{l}\mathrm{n}w
6 \mathrm{l}\mathrm{n}w2
,
respectively.
Proof. Using Lemma 3.2 and the fact that | f \prime | is monotonically decreasing and geometrically
relative convex function, we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1
6
\biggl[
f(g(a)) + 4f
\biggl(
g(a) + g(b)
2
\biggr)
+ f(g(b))
\biggr]
- 1
g(b) - g(a)
g(b)\int
g(a)
f(g(x))dg(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq (g(b) - g(a))
\left\{
1
2\int
0
\bigm| \bigm| \bigm| \bigm| t - 1
6
\bigm| \bigm| \bigm| \bigm| | f \prime ((1 - t)g(a) + tg(b))| dt +
+
1\int
1
2
\bigm| \bigm| \bigm| \bigm| 16 - t
\bigm| \bigm| \bigm| \bigm| | f \prime ((1 - t)g(a) + tg(b))| dt
\right\} \leq
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
SIMPSON-TYPE INEQUALITIES FOR GEOMETRICALLY RELATIVE CONVEX FUNCTIONS 995
\leq (g(b) - g(a))
\left\{
1
2\int
0
\bigm| \bigm| \bigm| \bigm| t - 1
6
\bigm| \bigm| \bigm| \bigm| | f \prime ((g(a))1 - t(g(b)t))| dt +
+
1\int
1
2
\bigm| \bigm| \bigm| \bigm| 16 - t
\bigm| \bigm| \bigm| \bigm| | f \prime ((g(a))1 - t(g(b)t))| dt
\right\} \leq
\leq (g(b) - g(a))| f \prime (g(a))|
\left\{
1
2\int
0
\bigm| \bigm| \bigm| \bigm| t - 1
6
\bigm| \bigm| \bigm| \bigm| \biggl( | f \prime (g(b))|
| f \prime (g(a))|
\biggr) t
dt+
1\int
1
2
\bigm| \bigm| \bigm| \bigm| 16 - t
\bigm| \bigm| \bigm| \bigm| \biggl( | f \prime (g(b))|
| f \prime (g(a))|
\biggr) t
dt
\right\} =
= (g(b) - g(a))| f \prime (g(a))|
\left\{
1
2\int
0
\bigm| \bigm| \bigm| \bigm| t - 1
6
\bigm| \bigm| \bigm| \bigm| wtdt+
1\int
1
2
\bigm| \bigm| \bigm| \bigm| 16 - t
\bigm| \bigm| \bigm| \bigm| wtdt
\right\} =
= (g(b) - g(a))| f \prime (g(a))| \{ F1(t, w) + F2(t, w)\} .
Theorem 3.1 is proved.
Theorem 3.2. Let f : I \subset \BbbR \rightarrow \BbbR be a differentiable function on I\circ where g(a), g(b) \in I with
g(a) < g(b). If f \prime \in L[g(a), g(b)] and | f \prime | q is monotonically decreasing and geometrically relative
convex function, then for
1
p
+
1
q
= 1 following inequality holds:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1
6
\biggl[
f(g(a)) + 4f
\biggl(
g(a) + g(b)
2
\biggr)
+ f(g(b))
\biggr]
- 1
g(b) - g(a)
g(b)\int
g(a)
f(g(x))dg(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq (g(b) - g(a))| f \prime (g(a))|
\biggl(
1 + 2p+1
6p+1(p+ 1)
\biggr) 1
p \Bigl[
(H1(t, q, w))
1
q + (H2(t, q, w))
1
q
\Bigr]
,
where
H1(t, q, w) =
1
2\int
0
wqtdt,
and
H2(t, q, w) =
1\int
1
2
wqtdt,
respectively.
Proof. Using Lemma 3.2, Hölder’s inequality and the fact that | f \prime | q is monotonically decreasing
and geometrically relative convex function, we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1
6
\biggl[
f(g(a)) + 4f
\biggl(
g(a) + g(b)
2
\biggr)
+ f(g(b))
\biggr]
- 1
g(b) - g(a)
g(b)\int
g(a)
f(g(x))dg(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
996 M. A. NOOR, K. I. NOOR, M. U. AWAN
\leq (g(b) - g(a))
\left\{
\left(
1
2\int
0
\bigm| \bigm| \bigm| \bigm| t - 1
6
\bigm| \bigm| \bigm| \bigm| p dt
\right)
1
p
\left(
1
2\int
0
| f \prime ((1 - t)g(a) + tg(b))| qdt
\right)
1
q
+
+
\left( 1\int
1
2
\bigm| \bigm| \bigm| \bigm| t - 5
6
\bigm| \bigm| \bigm| \bigm| p dt
\right)
1
p
\left( 1\int
1
2
| f \prime ((1 - t)g(a) + tg(b))| qdt
\right)
1
q
\right\} \leq
\leq (g(b) - g(a))| f \prime (g(a))|
\left\{
\left(
1
2\int
0
\bigm| \bigm| \bigm| \bigm| t - 1
6
\bigm| \bigm| \bigm| \bigm| p dt
\right)
1
p
\left(
1
2\int
0
\biggl(
| f \prime (g(b))|
| f \prime (g(a))|
\biggr) qt
dt
\right)
1
q
+
+
\left( 1\int
1
2
\bigm| \bigm| \bigm| \bigm| t - 5
6
\bigm| \bigm| \bigm| \bigm| p dt
\right)
1
p
\left( 1\int
1
2
\biggl(
| f \prime (g(b))|
| f \prime (g(a))|
\biggr) qt
dt
\right)
1
q
\right\} \leq
\leq (g(b) - g(a))| f \prime (g(a))|
\biggl(
1 + 2p+1
6p+1(p+ 1)
\biggr) 1
p \Bigl[
(H1(t, q, w))
1
q + (H2(t, q, w))
1
q
\Bigr]
.
Theorem 3.2 is proved.
Theorem 3.3. Let f : I \subset \BbbR \rightarrow \BbbR be a differentiable function on I\circ , where g(a), g(b) \in I with
g(a) < g(b). If f \prime \in L[g(a), g(b)] and | f \prime | q is monotonically decreasing and geometrically relative
convex function, then for
1
p
+
1
q
= 1 following inequality holds:
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1
6
\biggl[
f(g(a)) + 4f
\biggl(
g(a) + g(b)
2
\biggr)
+ f(g(b))
\biggr]
- 1
g(b) - g(a)
g(b)\int
g(a)
f(g(x))dg(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq (g(b) - g(a))| f \prime (g(a))|
\biggl(
2(1 + 2p+1)
6p+1(p+ 1)
\biggr) 1
p \Bigl[
(H(t, q, w))
1
q
\Bigr]
,
where
H(t, q, w) =
1\int
0
wqtdt.
Proof. Using Lemma 3.2, Hölder’s inequality and the fact that | f \prime | q is monotonically decreasing
and geometrically relative convex function, we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1
6
\biggl[
f(g(a)) + 4f
\biggl(
g(a) + g(b)
2
\biggr)
+ f(g(b))
\biggr]
- 1
g(b) - g(a)
g(b)\int
g(a)
f(g(x))dg(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
SIMPSON-TYPE INEQUALITIES FOR GEOMETRICALLY RELATIVE CONVEX FUNCTIONS 997
\leq (g(b) - g(a))
\left( 1\int
0
| \mu (t)| p dt
\right)
1
p
\left( 1\int
0
| f \prime ((1 - t)g(a) + tg(b))| qdt
\right)
1
q
\leq
\leq (g(b) - g(a))| f \prime (g(a))|
\left(
1
2\int
0
\bigm| \bigm| \bigm| \bigm| t - 1
6
\bigm| \bigm| \bigm| \bigm| p dt+
1\int
1
2
\bigm| \bigm| \bigm| \bigm| t - 5
6
\bigm| \bigm| \bigm| \bigm| p dt
\right)
1
p \left( 1\int
0
\biggl(
| f \prime (g(b))|
| f \prime (g(a))|
\biggr) qt
dt
\right)
1
q
=
= (g(b) - g(a))| f \prime (g(a))|
\biggl(
2(1 + 2p+1)
6p+1(p+ 1)
\biggr) 1
p \Bigl[
(H(t, q, w))
1
q
\Bigr]
.
Theorem 3.3 is proved.
Theorem 3.4. Let f : I \subset \BbbR \rightarrow \BbbR be a differentiable function on I\circ , where g(a), g(b) \in I with
g(a) < g(b). If f \prime \in L[g(a), g(b)] and | f \prime | q is monotonically decreasing and geometrically relative
convex function, then for
1
p
+
1
q
= 1 following inequality holds:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1
6
\biggl[
f(g(a)) + 4f
\biggl(
g(a) + g(b)
2
\biggr)
+ f(g(b))
\biggr]
- 1
g(b) - g(a)
g(b)\int
g(a)
f(g(x))dg(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq (g(b) - g(a))| f \prime (g(a))|
\biggl(
2(1 + 2p+1)
6p+1(p+ 1)
\biggr) 1
p \Bigl[
(H(t, q, w))
1
q
\Bigr]
,
where
H(t, q, w) =
1\int
0
wqtdt.
Proof. Using Lemma 3.2, Holder’s inequality and the fact that | f \prime | q is monotonically decreasing
and geometrically relative convex function, we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1
6
\biggl[
f(g(a)) + 4f
\biggl(
g(a) + g(b)
2
\biggr)
+ f(g(b))
\biggr]
- 1
g(b) - g(a)
g(b)\int
g(a)
f(g(x))dg(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq (g(b) - g(a))
\left( 1\int
0
| \mu (t)| p dt
\right)
1
p
\left( 1\int
0
| f \prime ((1 - t)g(a) + tg(b))| qdt
\right)
1
q
\leq
\leq (g(b) - g(a))| f \prime (g(a))|
\left(
1
2\int
0
\bigm| \bigm| \bigm| \bigm| t - 1
6
\bigm| \bigm| \bigm| \bigm| p dt+
1\int
1
2
\bigm| \bigm| \bigm| \bigm| t - 5
6
\bigm| \bigm| \bigm| \bigm| p dt
\right)
1
p \left( 1\int
0
\biggl(
| f \prime (g(b))|
| f \prime (g(a))|
\biggr) qt
dt
\right)
1
q
=
= (g(b) - g(a))| f \prime (g(a))|
\biggl(
2(1 + 2p+1)
6p+1(p+ 1)
\biggr) 1
p \Bigl[
(H(t, q, w))
1
q
\Bigr]
.
Theorem 3.4 is proved.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
998 M. A. NOOR, K. I. NOOR, M. U. AWAN
Theorem 3.5. Let f : I \subset \BbbR \rightarrow \BbbR be a differentiable function on I\circ , where g(a), g(b) \in I with
g(a) < g(b). If f \prime \in L[g(a), g(b)] and | f \prime | q is monotonically decreasing and geometrically relative
convex function, then for
1
p
+
1
q
= 1 following inequality holds:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1
6
\biggl[
f(g(a)) + 4f
\biggl(
g(a) + g(b)
2
\biggr)
+ f(g(b))
\biggr]
- 1
g(b) - g(a)
g(b)\int
g(a)
f(g(x))dg(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq (g(b) - g(a))| f \prime (g(a))|
\Biggl(
2(q - 1)(1 + 2
2q - 3
q - 1 )
(2q - 3)6
2q - 3
q - 1
\Biggr) 1 - 1
q \Bigl[
(L(t, q, w))
1
q
\Bigr]
,
where
L(t, q, w) =
1\int
0
| \mu (t)| 2wqtdt.
Proof. Using Lemma 3.2, Hölder’s inequality and the fact that | f \prime | q is monotonically decreasing
and geometrically relative convex function, we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1
6
\biggl[
f(g(a)) + 4f
\biggl(
g(a) + g(b)
2
\biggr)
+ f(g(b))
\biggr]
- 1
g(b) - g(a)
g(b)\int
g(a)
f(g(x))dg(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq (g(b) - g(a))
\left( 1\int
0
| \mu (t)|
q - 2
q - 1 dt
\right) 1 - 1
q
\left( 1\int
0
| \mu (t)| 2 | f \prime ((1 - t)g(a) + tg(b))| qdt
\right)
1
q
\leq
\leq (g(b) - g(a))| f \prime (g(a))|
\left(
1
2\int
0
\bigm| \bigm| \bigm| \bigm| t - 1
6
\bigm| \bigm| \bigm| \bigm| q - 2
q - 1
dt+
1\int
1
2
\bigm| \bigm| \bigm| \bigm| t - 5
6
\bigm| \bigm| \bigm| \bigm| q - 2
q - 1
dt
\right)
1 - 1
q
\times
\times
\left( 1\int
0
| \mu (t)| 2
\biggl(
| f \prime (g(b))|
| f \prime (g(a))|
\biggr) qt
dt
\right)
1
q
=
= (g(b) - g(a))| f \prime (g(a))|
\Biggl(
2(q - 1)(1 + 2
2q - 3
q - 1 )
(2q - 3)6
2q - 3
q - 1
\Biggr) 1 - 1
q \Bigl[
(L(t, q, w))
1
q
\Bigr]
.
Theorem 3.5 is proved.
Theorem 3.6. Let f : I \subset \BbbR \rightarrow \BbbR be a differentiable function on I\circ , where g(a), g(b) \in I with
g(a) < g(b). If f \prime \in L[g(a), g(b)] and | f \prime | q is monotonically decreasing and geometrically relative
convex function, then for q > 1 following inequality holds:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1
6
\biggl[
f(g(a)) + 4f
\biggl(
g(a) + g(b)
2
\biggr)
+ f(g(b))
\biggr]
- 1
g(b) - g(a)
g(b)\int
g(a)
f(g(x))dg(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
SIMPSON-TYPE INEQUALITIES FOR GEOMETRICALLY RELATIVE CONVEX FUNCTIONS 999
\leq (g(b) - g(a))| f \prime (g(a))|
\biggl(
1 + 2p+1
6p+1(p+ 1)
\biggr) 1
p \Bigl[
(H1(t, q, w))
1
q + (H2(t, q, w))
1
q
\Bigr]
,
where
H1(t, q, w) =
1
2\int
0
wqtdt
and
H2(t, q, w) =
1\int
1
2
wqtdt,
respectively.
Proof. Using Lemma 3.2, power mean inequality and the fact that | f \prime | q is monotonically
decreasing and geometrically relative convex function, we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1
6
\biggl[
f(g(a)) + 4f
\biggl(
g(a) + g(b)
2
\biggr)
+ f(g(b))
\biggr]
- 1
g(b) - g(a)
g(b)\int
g(a)
f(g(x))dg(x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq (g(b) - g(a))
\left\{
\left(
1
2\int
0
\bigm| \bigm| \bigm| \bigm| t - 1
6
\bigm| \bigm| \bigm| \bigm| dt
\right)
1 - 1
q
\left(
1
2\int
0
\bigm| \bigm| \bigm| \bigm| t - 1
6
\bigm| \bigm| \bigm| \bigm| | f \prime ((1 - t)g(a) + tg(b))| qdt
\right)
1
q
+
+
\left( 1\int
1
2
\bigm| \bigm| \bigm| \bigm| t - 5
6
\bigm| \bigm| \bigm| \bigm| dt
\right)
1 - 1
q
\left( \bigm| \bigm| \bigm| \bigm| t - 5
6
\bigm| \bigm| \bigm| \bigm|
1\int
1
2
| f \prime ((1 - t)g(a) + tg(b))| qdt
\right)
1
q
\right\} \leq
\leq (g(b) - g(a))| f \prime (g(a))|
\left\{
\left(
1
2\int
0
\bigm| \bigm| \bigm| \bigm| t - 1
6
\bigm| \bigm| \bigm| \bigm| dt
\right)
1 - 1
q
\left(
1
2\int
0
\bigm| \bigm| \bigm| \bigm| t - 1
6
\bigm| \bigm| \bigm| \bigm| \biggl( | f \prime (g(b))|
| f \prime (g(a))|
\biggr) qt
dt
\right)
1
q
+
+
\left( 1\int
1
2
\bigm| \bigm| \bigm| \bigm| t - 5
6
\bigm| \bigm| \bigm| \bigm| dt
\right)
1 - 1
q
\left( 1\int
1
2
\bigm| \bigm| \bigm| \bigm| t - 5
6
\bigm| \bigm| \bigm| \bigm| \biggl( | f \prime (g(b))|
| f \prime (g(a))|
\biggr) qt
dt
\right)
1
q
\right\} \leq
\leq (g(b) - g(a))| f \prime (g(a))|
\biggl(
1 + 2p+1
6p+1(p+ 1)
\biggr) 1
p \Bigl[
(H1(t, q, w))
1
q + (H2(t, q, w))
1
q
\Bigr]
.
Theorem 3.6 is proved.
References
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ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
1000 M. A. NOOR, K. I. NOOR, M. U. AWAN
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Received 21.10.14
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 7
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| id | umjimathkievua-article-1612 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:09:07Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/6c/c9ac46ad1c0d8228104f75fb8625c16c.pdf |
| spelling | umjimathkievua-article-16122019-12-05T09:20:38Z Simpson-type inequalities for geometrically relative convex functions Нерiвностi типу сiмпсона для геометрично вiдносних опуклих функцiй Awan, M. U. Noor, K. I. Noor, M. A. Аван, М. У. Нур, К. І. Нур, М. А. We consider a class of geometrically relative convex functions and deduce several new integral inequalities of Simpson’s type via geometrically relative convex functions. The ideas and techniques used in the paper may stimulate further research in this area. Розглянуто клас геометрично вiдносних опуклих функцiй та отримано кiлька нових iнтегральних нерiвностей типу Сiмпсона в термiнах геометрично вiдносних опуклих функцiй. Iдеї i технiка, що використовуються в роботi, можуть стимулювати подальшi дослiдження в данiй областi. Institute of Mathematics, NAS of Ukraine 2018-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1612 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 7 (2018); 992-1000 Український математичний журнал; Том 70 № 7 (2018); 992-1000 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1612/594 Copyright (c) 2018 Awan M. U.; Noor K. I.; Noor M. A. |
| spellingShingle | Awan, M. U. Noor, K. I. Noor, M. A. Аван, М. У. Нур, К. І. Нур, М. А. Simpson-type inequalities for geometrically relative convex functions |
| title | Simpson-type inequalities for geometrically relative convex
functions |
| title_alt | Нерiвностi типу сiмпсона для геометрично вiдносних
опуклих функцiй |
| title_full | Simpson-type inequalities for geometrically relative convex
functions |
| title_fullStr | Simpson-type inequalities for geometrically relative convex
functions |
| title_full_unstemmed | Simpson-type inequalities for geometrically relative convex
functions |
| title_short | Simpson-type inequalities for geometrically relative convex
functions |
| title_sort | simpson-type inequalities for geometrically relative convex
functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1612 |
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